Usage and alternative words
A simple answer to this one: no, there is no word for pescetarians that is more commonly used or understood than pescetarian (or pescatarian, if you prefer that spelling).
It is not a concept that has been spoken about commonly for very long, and pescetarian is, to my knowledge, the only word for it that has any practical currency at all. It is not overly common, but it is also not limited to “strict circles”. It is not uncommon for average people (with no special dietary habits) to know at least more or less what a pescetarian is.
A perhaps more transparent alternative is pesco-vegetarian, which more overtly marks the connection between pescetarianism and vegetarianism. It may be easier for people who do not know what pescetarianism is to grasp the meaning of pesco-vegetarian, but it is not in as common use as pescetarian.
At least equally common as pescetarian, though, is misuse of the term vegetarian. By any commonly accepted definition, a vegetarian does not eat any kind of meat, including poultry and fish; but quite a few people do not consider fish and/or poultry ‘meat’ and thus do not distinguish vegetarians and pescetarians (and often not vegans, either).1 It is my personal experience, however, that the percentage of people who use vegetarian to refer to pescetarians is decreasing. Ten or 15 years ago, nearly everyone thought people who only ate fish were vegetarian; nowadays, the split between those who call it vegetarian and those who call it pescetarian is more like 50/50. (Excluding of course the ones who do not know that there is a term for either thing, which these days is a definite minority.)
Your best bet is to simply get used to pescetarian, even if you do think it is awkward, because there is no better alternative. Alternatively, if you want to make sure you’re understood and don’t care it’s a bit longer, explain: “I tend towards vegetarianism, but I do eat fish as well” or something like that.
Origin of the word
The established theory
As both Merriam-Webster, Wikipedia, and the OED mention, the formation of the word is not quite clear. The second element, -tarian, is clear enough: it originated in vegetarian, whence it spread to other types of ‘dietary lifestyles’. Vegetarian is itself a more or less ad-hoc blend of vegetable (or possibly vegetation) and the weakly productive suffix -arius.
The first element is more difficult to track down. It is quite obviously the word found in Latin piscis and Italian pesce, both meaning (and cognate to) ‘fish’—but its form and pronunciation doesn’t really match anything.
Italian pesce has the /e/ of pescetarian, but ⟨sc⟩ before a(n orthographic) front vowel represents /ʃ/ in Italian, and pesce is phonetically [ˈpeʃe]. This is different from English where pescetarian is generally pronounced /pεskɨtεəriən/2, with /sk/.
In Classical Latin, ⟨c⟩ always represented /k/, and piscis was phonemically and phonetically [ˈpiskis]; in later Latin as well as in English words derived from the Latin word, however, ⟨sc⟩ represented first the /ʃ/ of Italian and later on (in French, where English got most of her Latin words from) a simple /s/. Thus for example the Zodiac sign Pisces is /ˈpaɪsiːz/ in English, not */ˈpɪskiːz/ or anything like that. On the other hand, there are English words derived from the Latin word where the ⟨sc⟩ appears before a back vowel, in which case the pronunciation never changed and is still /sk/ in English. There are such words in Italian as well, where the Italian /e/ is maintained—but not in English.
The OED has a total of 43 words beginning with pisc- (excluding two for the liquor known as pisco (Collins), which is named after a Peruvian city whose name, according to Wiktionary, is from Quechua and thus unrelated). Of these 43 words, 26 begin with pisce- or pisci- and thus have no /k/ sound, while 16 begin with pisca- (14), pisco- (1), or piscu- (1). The pisco- and piscu- words (piscose and pisculent) are both marked as obsolete as are two of the pisca-, so basically, there are a handful of words beginning in pisca-, all pronounced /ˈpɪskə-/.
Contrarily, there is only one word in the entire OED beginning in pesc-, and that is pescetarian.
Both Merriam-Webster and the OED agree that in pescetarian, the first element is “probably” taken from Italian, but they cannot really explain why the word has /sk/ in English. The OED article says:
If the assumption that the word was formed on Italian pesce is correct, the pronunciation with /sk/ (rather than /ʃ/ ) and the spelling pescatarian both suggest that this origin has been opaque to many from an early date.
An alternative theory
Personally, I would suggest a slightly different scenario:
I believe the word was originally coined not from Italian, but from Latin, and I would hypothesise that whoever first came up with it in the late 1980s or early 1990s actually started out with *piscatarian, to go with adjectives like piscatorial (‘of or concerning fishermen or fishing’). Since the coinage was so obviously meant to be a ‘spinoff’ of vegetarian, though, the vowel structure of piscatarian was adapted to fit that of vegetarian; thus (showing only the vowels with “.” as the syllable separator):
piscatarian /ɪ.ə.ˈεə.iə/ ⇒ /ε.ə.ˈεə.iə/
Once the word started gaining popularity and entering wider usage, it was already pescatarian, and the link to the Latin root pisc- was somewhat tenuous and strained. Tenuous enough, at least, that people with limited knowledge of Italian started connecting the word to the Italian word pesce (often seen on menus in Italian restaurants) instead of the Latin root, and also began spelling it like the Italian word.3
The hypothetically ‘original’ spelling piscatarian (as well as piscetarian, of uncertain pronunciation) have appeared here and there in the early days; but they are far less common than the forms with ⟨e⟩.
As for why one would prefer one spelling over another—well, that’s just personal preference. I personally prefer pescetarian because it maintains the same vowel skeleton as vegetarian not only phonologically, but also orthographically (and preferring in general British orthography, words like sceptic mean that I don’t find ⟨sce⟩ pronounced /skε/ all that unacceptable). One may equally prefer pescatarian because it is more intuitive for the ⟨c⟩ to represent /k/ before ⟨a⟩ than before ⟨e⟩ (especially if one generally prefers US orthography, where skeptic is not a parallel for ⟨sce⟩ as /skε/).
There is an awful lot of variation between who considers what ‘vegetarian’ and ‘non-vegetarian’. At one end of the spectrum are those who cling to the very narrowest possible definition of vegetarianism: someone who does everything they can to avoid eating, consuming, or otherwise using any part of an animal whose extraction directly or indirectly impacts the animal negatively: meat, skin, leather, ivory, etc. At the opposite end are those who call themselves vegetarian because they don’t eat red meat but do eat pork, poultry, fish, seafood, etc. I would say there is some leeway in how strict you need to go (I have no problem with calling someone who uses regular soap or wears leather shoes a vegetarian), but once your diet starts including the meat from animals, there’s little point left in terming it ‘vegetarianism’ anymore.
I’m using Wikipedia’s IPA notation for English to indicate phonemic writing here.
I imagine someone with proper Italian skills would not have done this, since they would be familiar with other Italian words that have [pesk-], like the verb pescare [pesˈkaːɾe] ‘to fish’. Assuming pescetarian is an urban American invention, though, most people who might use it are relatively likely to know pesce from restaurants, but less likely to know any other Italian words from the same root.