Hi Pedroski,
It's been decades since I "diagrammed a sentence." God knows we spent endless hours in the 4th & 5th grade doing this. Looking back, I think it was all just a clever ploy invented by nuns to keep their charges tamed.
(By the way, having just seen the movie Doubt, starring Meryl Streep and Philip Seymour Hoffman, I can't recommend it enough. Anyway, in the background of one of the classroom scenes is a green schoolhouse "blackboard" with an intricately diagrammed sentence. - it was deja vu all over again! Yogi Berra must've diagrammed more than his share of sentences in his time. And look how he turned out!
)
Yes, about your question ...

Originally Posted by
Pedroski ... or is that short sentence an ellipsis, and I should really be writing: That eel is Tom's eel. Which makes Tom's a determiner.
Yes, you're on the right track. The sentence is certainly an ellipsis. In the Romance (or Latin) languages, there is no construction that is equivalent to the English possessive as designated by an "apostrophe - s" (" 's "). Instead, these languages follow the construction of the Vulgar Latin possessive case -- which English may also do: ".. the eel of Tom" or "...the eel of Bob". Thus, the ellipsis of your sentence may be expanded to:
That eel is not the eel of Tom, but (it is) the eel of Bob.
- or an even greater expansion:
That eel is not the eel that belongs to Tom, but the eel that belongs to Bob.
In any event, it should now be clear that in your sentence both Bob's and Tom's are possessive case nouns.
As to how one would diagram the sentence? - Please! I don't want to know!