I think maybe the right term here is not "unfalsifiable" but David Deutsch's "explaining too much". https://en.wikipedia.org/wiki/Explanatory_power#Overview

But if you do post about it (perhaps you're hoping to find like-minded gardeners in your community?) and I say you're just doing it to signal, how would you falsify that?

Isn’t sharing it publicly already a counter argument to “I don’t care if nobody knows about it”? Thus you wouldn’t have to falsify.

We could think of a scenario where his neighbor shares information about his (the authors) garden. In this case, if the author would seek measures to delete these posts, it might be interpreted as “signaling that he does not care and even doesn’t want others to know”.

I agree with you that signaling always shows intention (ie wanting something). You can “want not to want something”, though (:

I think if you define "signaling" as "wanting someone to know about it" then of course any social media communication is "signaling," but then is any communication at all. Oviously when I speak I want to be head, so it's not "I don't care if nobody knows about it."

But I think that's far too broad a definition of "signaling." The original article says "Building social capital through signaling."

When you post about your gardening to see if there are any other gardeners you can connect with, I don't think it's necessarily to "build your social capital," yet you can prove that it isn't.