First, neither word is at the end of the sentence there: “basketball” and “the students” come after. It doesn’t make sense to call that the end of the sentence.
Secondly, what makes this use of “or” and “but” special here is that they don’t have a full clause with them: only a few words come after the conjunction.
Here two full clauses are joined by “or”. A (full) clause is a phrase that contains a finite verb (like “will” here).
Here “or” joins two phrases that are not clauses (“Cleopatra” and “Marc Anthony”), as in your example. This is quite common, especially with coordinating conjunctions (conjunctions that join two parallel clauses or phrases, like “and”, “or”, and “but”).
But it can also occur with subordinating conjunctions:
Note that you can normally only use a phrase/fragment ( = not a full clause) if you can copy the missing information from the main clause: here it is only possible because the subject (“Cleopatra”) of the subordinate clause/fragment is the same as that of the main clause, and because the finite verb “was” can easily be added.
With coordinate conjunctions, it is less clear that you can copy the missing information to form two full clauses:
This is easy enough: I like cricket and [I like] football.
Here I can’t really expand the phrase after “or” into a full clause. That is because of “raising”, a special grammatical process that you can look up in Wikipedia. If I had to expand the sentence into full clauses, I would have to do it like this:
But this is a bit far fetched. It is easiest to remember that most coordinate conjunctions can easily be used with phrases that aren’t clauses.
That said, “but” is a bit special, because its meaning changes a bit when it is not used with a full clause but with a phrase/fragment as in your example; then it means something like “only”:
Here it means “only”, and this use of “but” is rather old fashioned.
With a negation (like “no”, “not”, “nothing”, “no-one”, etc.), “but” is still quite common in this way; it is not very old fashioned, though it is certainly not informal. It works like a preposition here: it means “except”. That is why your dictionary said it was not really a conjunction when used in this way, and I see; I just think that their “it is used at the end of the sentence” is not a very good clue.