What is the correct pronunciation of a word if it sounds so much different in so many different accents?

It has more than one acceptable pronunciation. There is nothing that makes a pronunciation inherently "correct" beyond it being used by native speakers. Many English words have more than one pronunciation that is in common use by native speakers: either, tomato, route, bath, etc.

The role of a dictionary is to be descriptive, not prescriptive, that is to say, their role is to describe what English speakers say, not prescribe what they should say.

I know that the words there, their, and they're are homophones

Yes.

but I can't help but think that their has a slight /j/ between /e/ and /r/ (/ðer/).

It’s possible, but no more so than for there and they’re. I don’t think you’re imagining things, but I also wouldn’t recommend taking any special efforts to pronounce this word this way.

For American English speakers, there is generally no phonemic contrast between the vowel in there/their/they’re and the vowel in they/say/fade/male/mail. (This holds regardless of whether we’re talking about an accent with or without the Mary-marry-merry merger.)

But phonetically, you’re less likely to hear a /j/-like glide when the vowel is followed by /r/ with no intervening word boundary, and more likely to hear a glide when the vowel is word-final, followed by a vowel, or followed by most other consonants, like /p b m t d n k g s z f v θ ð tʃ dʒ/. For me, the /j/-like off glide of the vowel /e/ is also suppressed to some extent before a following /l/ (which I pronounce as “dark l”, [ɫ]), but I don’t know how common this is or the details of how it works (I do distinguish pairs like fail and fell).