What precisely is infinite ambiguity in a grammar?
From a comment by Sylvain:

A grammar is infinitely ambiguous iff there is a productive and accessible nonterminal $A$ s.t. $A \Rightarrow^+ A$. Nonterminals $A$ and $B$ fulfill this characterization in your first example, and so does $S$ in the second example when you add the rule $S \to \varepsilon$. I'll leave the proof of the characterization as an exercise.


   Consider that a parse tree of infinite size for a finite input sentence must repeat a nonterminal spanning the same interval in the input.