As an independent, individual programmer:
How do I let people very quickly know that I have not abandoned the software I've written and given away for free? That I am putting in the effort required to maintain and support my software to a professional level?
When software written by one or two developers is available for free, or marked as open-source, usually the default assumption is that it's abandon-ware. This is usually a safe assumption - check out the answers to this question if you doubt it: Why do programmers write closed source applications and then make them free?.
There are lots of programmers who provide free and/or open-source tools which are not abandon-ware, though.
If we're talking about large companies, ie Google, there's no real problem telling the difference between supported, live tools and software, and those which are abandoned or discontinued.
A lively git repository isn't quick - users will have to be savvy enough to understand the repository and know where to look for it. Consistent marketing and community management take more time and effort than I can put in on my own. Also, if my software becomes popular/successful, I assume those will grow on their own, and be supported by power users in the community.