Disambiguation of AC Analysis and Small Signal Analysis
Yes, when we say "AC analysis" we mean a small-signal analysis of the variations around an operating point due to an AC stimulus.

This meaning has been reinforced over the past 40ish years by the widely used SPICE simulation program, where a punch card or line in an input file that begins with .AC is used to initiate a small-signal AC analysis.

As you say, we can also do a small signal analysis to determine the effect of a small dc perturbation about an operating point. With that in mind, the term "small-signal analysis" is somewhat broader than "AC analysis".