Many people would say the correct form is "If I were rich ...".
In modern colloquial English though most younger people would say "If I was rich ...".
Prescriptivists might say the latter is "the subjunctive mood" and the former is "just plain wrong".
Both forms "were" and "was" are identical to English past tense forms. There are no distinct subjunctive forms I am aware of in Modern English though I do not know if Old English had them.
When learning foreign languages such as Spanish or German we are often told "English doesn't have the subjunctive". Such languages usually do have distinct forms for the subjunctive.
So my question is what terms are used in English linguistics to cover each case? Is it just "subjunctive" or are there also now terms such as "colloquial subjunctive", "informal subjunctive", "formal subjunctive"? Or are there people who insist that English is unlike Spanish and German and name these constructions something distinct from "subjunctive"? Also how do the terms "counterfactual" and "irrealis" fit in?
I didn't find this related question when I was writing mine: Why have the subjunctive and indicative converged in Modern English?
After much Googling I've noticed the phrase "marked subjunctive" seems to have some currency but I haven't yet seen it specifically defined to mean the use of "were" rather than "was" so the hunt continues but it's the closest so far.