This area of English, generally, may be indeed more appropriate for the ELL site, but this question does regard a part of linguistics that tends to fly under the radar of EFL students below C1 level (Cambridge).
But, first off, the title is misleading. Clauses are of no consequence here. Just the same as having the utterance It reminded me of the/a house I rented in Oxford., we could've had It reminded me of the/an Oxford rented house of mine., which features no additional clause. Anyway...
Pragmatics is the concerned area, and—unlike syntax, which studies sentence patterns, and semantics, which studies sentence meaning without context—pragmatics is a study of language use with respect to speaker’s communication intentions and its effects. In the example we have two speakers of the same or similar cultural background (as evidenced by their particular names) casually conversing about dwellings of the type common to their culture and located at likewise familiar locations. As the readers, we detect these similarities, for these native speakers of the English language are not tattling about Chinese cave dwellings, Persian yurt tents or Chilean stilt houses; their subject is the common dwelling of karls, johns and sarahs at locations well known to us as countryside and Oxford. Also, we notice that John brings up his house immediately after having painted in Sarah's mind (by relying on their common cultural background and on Sarah's familiarity with the buyer of that house, as evidenced by John's referring to him only by his name) a picture of a similar house. Are similarities that important for the usage of articles that they warrant this many words?
A reminder is in place, that only some nouns (more correctly, phrases and nominal groups), either commonly known or belonging to a commonly-known class, have a fixed rule about the, a or zero article. Examples of the top of my head: what a fool, The Day of the Lord, National Geographic (The Economist is an exception), the River Thames, an auto mechanic (profession) etc.
In all other cases, the decision of the speaker of a conversation about which article to use hinges exclusively on his or her estimation of his or her listener's knowledge/expectations about the constituent elements of the message that the speaker is ready to send. A well made estimation secures high chances that the listener will correctly receive the intended message. The situation in which the communication takes place—is what enables the speaker to achieve such estimation. Situational contingency does not mean there aren't fixed grammar rules to which the speaker needs to adhere. There are! Most of them are fixed. The matter is, simply, that grammar rules were so made, back when, that they TAKE the situation INTO ACCOUNT.
Only sometimes does a grammar rule give the speaker enough leeway to form his message this way or that way without any sacrifice to the meaning. For example, there is a grammar rule about the order of the adjectives of different semantic priority (already formulated, by various grammarians), and there are same rules about the phonological priority, and even the sociolinguistic one, that are more hidden (and perhaps not yet overtly formulated, but nonetheless known to native speakers subconsciously), yet when the adjectives that are of the same priority in all regards are to be put together, all that English grammar says to the speakers is: It is up to you. (The utterance My gray and brown scarf remains identical to the utterance my brown and gray scarf if that scarf is in equal parts gray and brown.)
The choice of the article is NOT governed by the grammar rules so fixed that they can afford to be ignorant of the situation. (This is one aspect of English that is such a thorn in the side of many native speakers of the languages that don't use articles, and even of some speakers of the languages that do.)
Hereupon... The situation is what determines which article John MUST use. John and Sarah know what the situation is. 'Dunno', the asker of this question, is the third personage who does, for he had the whole plot and mise-en-scene in his mind before he converted that into words. But, we, the readers, do not comprehend the exact situation, as this text is too skimpy on the pieces of information crucial for the usage of articles. We can merely conjure up several versions, but for each of them, only one of the articles is fitting, and that article is prescribed by grammar. Some situations can be imagined more plausibly, some less. That falls under the jurisdiction of pragmatics. My take on some of the probable situations:
A. John was renting only one house while in Oxford, as he was a student there. (Why is that my first concept? Because, the notion of that city is strongly marked by one historic University of Oxford. The number of houses John rented is one, because Oxford students often come from upper classes, and their sons, perhaps because they feel otherwise would be beneath their rank, tend not to get on bad terms with landlords, and so the prospects of their being kicked out of a rented house, thus of being forced to rent again, are low. Also, Oxford students don't start out by renting some seedy place, requiring them to rent elsewhere, for they from the start entertain high hopes of their student years.) Sarah and John know each other rather well. (She both greets him with "hey" instead of the more formal hi and has heard from him personally about his private affair of plannig to go on a trip, otherwise she'd refer to his trip by mentioning its destination.) Therefore, Sarah already knows about John's only rented house in Oxford. John's aware of her knowing and doesn't want to pretend he isn't, or Dunno would've hinted at that, as such a behavior is a bit out of the ordinary. For the situation described, English grammar DEMANDS that John use the, and not a. There's no two ways about it. In such a situation, if John said It reminded me of a house I rented in Oxford., Sarah would quip Tut, like I don't know that you rented a house there.
B. John rented more than one house in Oxford. It's not implausible. First, because not all Oxford students are on good terms with their landlords, e.g., because both the student and the landlord may turn out to be just too obnoxious, after which they rid themselves of one another. Second, because an Oxford student may happen to be so rich that he rent one house for studying and one for dolce vita. Or, John was never an Oxford student. In Oxford, he worked. He perhaps didn't have the means to get into Oxford, or hadn't even ever planned to. After all, his name is quite class-neutral; it's not something like Sinclair or Prescott. It's quite plausible that, having had a real job, he wanted a different, better house once he got better-off. Everything else is the same as in A. For the situation described, English grammar DEMANDS that John use a, and not the.
C. Let everything remain the same as in B, except this: let us imagine John was nostalgic for his Oxford life. That's not out of the ordinary at all; Oxford is a beautiful place to live. As he mentions his Oxford house, and it doesn't matter whether it's his only rented one or but one of them, he goes into a sort of a passing reverie: he glances away from Sarah, squints and makes a pause. Sarah, the friend that she is, follows his suit, as friends do when they chat. During that pause, she has conjured up in her mind the house that John is conjuring up. Their visions may be quite different, but both of those visions represent "John's house that he rented in Oxford." Now, the key is that right as he was about to bring up his Oxford house—something that he, the nostalgic sap that he is—knew, by now, would put him into a state of reverie, John had predicted, since he knows her rather well, that Sarah would join him in the reverie, as she indeed did. So, soon they will both think of "John's house that he rented in Oxford" and John knows it. Let's assume that he also likes it; that he likes his good connection with Sarah. And that he doesn't want to do anything to undermine it. For the situation described, English grammar DEMANDS that John use the, and not a.
D. Let us leave everything the same as in C, except let's assume that John doesn't appraise too highly his bond with Sarah. Perhaps he's never got over something she once did. It's not implausible; many friendships conceal some grudges. As he senses that a mutual moment of reverie is approaching, he refuses to acknowledge that he senses it, because that's just the limit of their friendship he does not want to cross. For the situation described, English grammar DEMANDS that John use a, and not the.
E. John worked in Oxford for only several months, renting only one house there and never enrolling in any universities, and he has worked and rented a house in many places. Due to this, Sarah happens not to know about his Oxford past. They have a mutual friend, Karl, who is a conformer, a chap rather bland and predictable. Therefore, Sarah, being the archetypal girl-with-an-English-name that she is, easily surmises what Karl's new countryside home, that John has just referenced, might look like. Perhaps she's even seen it already. It makes no difference; John concludes that Sarah has visualized Karl's house just now. Also, he is proud of having lived and worked in many places, so, as he gets ready to bring up his Oxford house that is similar to Karl's, and the fact that he lived in that nice city, he doesn't want Sarah's reaction to be, Oh, you've lived in Oxford?, but rather more, Of course, you've lived in Oxford too. It figures. (—Of course.) When he references that Oxford house, he wants Sarah to visualize it directly (without any surprise moments delaying her), almost to the point of saying to him, Sure, I see it. And she really will see it, because it's similar to Frank's house, so there's no fear of Sarah resenting him for his expecting her to play along, and thus, their friendship will still wear well. For the situation described, English grammar DEMANDS that John use the, and not a.
The dialogue Dunno has given allows for countless other precisely defined situations aside from these five.
Everything having been considered, Dunno's question makes no sense. It is not some grammar rule that decides which article John must use in the dialogue Dunno's provided; it is John himself who decides, depending on how he wants to go about his relationship to his listener, Sarah. Ultimately, it is up to Dunno: he must visualize his literary characters in finer detail, and he will know which article to use. However, for whatever he visualizes, there are grammar rules (which this site does not aim to tutor on).
P. S. Bit of a TL/DR, ey? Sorry.