I've had two different students in two different classes correct my pronunciation of 'satyr' this term, which is interesting. I pronounce it SATTER (usually -- since I have some difficulty articulating words, I will often shift around in pronunciation even of ordinary words, and don't always know myself how it ended up being pronounced). They insisted it was SAYTER. According to the dictionaries, something like SATTER is the 'chiefly British' pronunciation, while SAYTER is the American pronunciation, which I find interesting. SAYTER is not especially close to the original Greek at all, and I can literally count on one hand the Americans I have ever heard pronounce the word with a long A, despite having talked with people about Greek mythology and culture a lot over the years. The natural way Americans tend to pronounce it when they come across the word without having ever heard it before (and it is definitely a word people are more likely to come across in reading than in conversation) is, as far as I can tell, SATIRE, which is an entirely reasonable guess and pronunciation, closer to the actual Greek than SAYTER, closer to the pronunciation of the adjectival form 'satyric' and 'satyrical' (which are homophonous with 'satiric' and 'satirical', the adjectival forms of 'satire'), and overwhelmingly the most common pronunciation I've actually come across.
Likewise, satyrid, the name for a family of butterflies and derived directly from the same origin, has three different pronunciations in most dictionaries -- SAYTERID, SATTERID, and SA-TIRE-ID. But the homophonic-with-satire pronunciation for 'satyr' doesn't seem to have been picked up on by dictionaries. One wonders, first, how SAYTER ever got purchase in the first place, and second, how it was determined that Americans pronounce 'satyr' this way.
Of course, whenever anyone corrects my pronunciation in any context, I always say, "People pronounce that word in many different ways," which is always undeniably true.