Defining an acid-base reaction from the perspective of the Arrhenius theory of acids and bases
I understand that ammonia counts as a base because it can react with water to give hydroxide ions, and HCl is an acid.

This is where you went wrong.
The Arrhenius theory is the oldest of the common acid-base theories. According to Arrhenius:

An acid is a compound that will dissociate in aquaeous solution forming $\ce{H+}$ ions.
A base is a compound that will dissociate in aquaeous solution forming $\ce{OH-}$ ions.

As per the Arrhenius theory, ammonia is not a base. Also, contrary to all later theories (Brønsted-Lowry, Lewis and Pearson), Arrhenius did not account for any acid-base reactions; instead, he intended to define only those substances whose solutions reacted acidicly or basicly. It is therefore meaningless to define an acid-base reaction according to the Arrhenius theory.
What Arrhenius did account for was neutralisation reactions. Indeed, if an Arrhenius acid and an Arrhenius base are mixed, the form water and a salt, which can be easily shown in the equation below:
$$\ce{(H+ + A^-) + (B+ + OH^-) -&gt; HOH + A^+B^-}$$
But it was not until Brønsted’s and Lowry’s theory that it was realised how a reaction should always be acid-base and not one or the other.