I think the words to use are "plaintiff's side civil litigation [as opposed to defense litigation], specializing in civil rights, with some pro bono criminal defense." That's specific enough that you can start a conversation with someone, anyway, and I think it's what you're talking about.
At minimum, you've established that you want to be a litigator, rather than, say, a transactional lawyer or something. You also don't want to be, e.g., a government prosecutor. So you start by looking at litigation departments at firms and seeing what they have that's similar to what you want to do. Granted, it's still a pretty wide field, but whoever told you that "everyone does that" either misunderstood you or is stupid.
That sounds like what I was looking for. I was thinking civil litigator but those usually apply laws tospecific situations. But, I also wanted to argue constitutionality of laws, like the work in the ACLU, but also do criminal cases.