There are non-Turing complete programming languages, and there are many things that are Turing complete but have nothing to do with programming (even PowerPoint), so this is neither a required nor sufficient property.

I believe a reasonable way to categorize languages as programming or not is simply.. what is it's primary use case. HTML's last two letters tell us exactly that it is not a programming language.

not sure if that's a good criterion. There are also non Turing complete languages like core SQL or Rocq that definitely feel like programming.

Things can and often do feel like other things that they are not.