Is there another way than [ɜr] to pronounce the grapheme "or" in words like "world" in AmEng?
Solution 1:
I listened to the pronunciation of "world" at the reference you gave to The Free Dictionary. It sounds to me just like the way I say it, which is a stressed, rounded retroflex vowel. Unstressed, in "butter" for instance, it's unrounded. I don't hear a variation in quality that would justify transcribing it as a diphthong.
The fact that the change in roundness of the vowel r going from unstressed to stressed r matches the change going from unrounded syllable offset r (as in "carp") to rounded syllable onset r (as in "crop") is one of the facts that convinces me that the stress/unstressed distinction applies to consonants, as well as vowels, with syllable onset consonants stressed and syllable offset consonants unstressed.
I confess to being unfamiliar with the diacritic notation you're using for describing pronunciation. Is that from Kenyon and Knott? (Consider using brackets to enclose pronunciations, instead of slashes, since that is the convention usually followed in phonology.)