"HBr is donating a proton to LiOH": Why doesn't it produce LiOH2?
I think it would produce something like &quot;LiOH2,&quot;

You are missing the charge. It should be something like $\ce{LiOH2+}$, which is an unconventional way of writing $\ce{Li+(aq) + H2O(l)}$.

So where is my misunderstanding?

You have to get used to interpreting the (aq) designation after ionic compounds. To help you out, here is the equation written with the ions shown as individual species (more typing, but closer to the actual situation):
$$\ce{Li+(aq) + OH-(aq) + H+(aq) + Br-(aq) -&gt; Li+(aq) + H2O(l) + Br-(aq)}$$