Conflicting definitions regarding TM and Recursively Enumerable languages
I wouldn't say that one definition has "prevailed".  If you look at the literature and textbooks, you'll see a large variety of definitions.

Different definitions might be useful in different contexts.  For example, if you're actually trying to program up a TM (for whatever reason) then the more expressive they are, the easier to program.  On the other hand, if you're trying to prove a lower bound or uncomputability result, picking the weakest TM might make life easier.  And the busy beaver function, for example, is usually defined relative to a fairly weak model.

I'd just recommend you use any model you feel comfortable with and tell the reader of your paper that is what you are using.