Real simple: if a series is promoted as a prequel, it's the "past". If it's not, it's the "present". And live-action trumps animation in my view.
But by that logic, DIS would be the "present" since it's not a prequel anymore. And SNW is both a sequel to DIS S2 and a prequel to TOS S1. And I don't think live-action "trumps" animation -- they're just as legitimate of storytelling mediums, and PRO and LD are both covering timeframes no one else has covered in settings no one else is covering.
I just think it's silly to hold onto the idea that there's any one "present" in Trek today, just like Star Trek V: The Final Frontier's 2286 was no more the "present" than TNG S3's 2366 was in 1989. This isn't the Berman era; there is no one "present." Star Trek is not a single continuing narrative; it's a collection of narratives with a common setting.
Edited to add:
DIS is also still generally considered to be the "flagship" ST series. So if how the shows are promoted determines when the "present" is, it would make more sense for 3190 to be the "present." But, again, I don't think promos should determine this -- Star Trek is not a single continuing narrative.