Maybe it is both? In one of the H2 cutscenes where Lucas Grey is in the asylum, his Providence board shows up for a moment and if you look closely you can see that Lord Beldingford and Alvaro are along with the other “main” members of Providence (on the right side of the picture). It’s the same pics that Ibbe posted above, even, just in black and white.
If Beldingford and Alvaro got included in the same board as the Partners, the Constant, and Soders themselves, then it’s a good enough indicator of them being Providence agents all along. Same thing could also go for Hayamoto. Perhaps when Lucas Grey investigated 47’s past targets, he found some of them were secretly Providence members, so that’s why they were included in his board in Colorado.
I think that’s a given. Not only do they have the same last name but I’m pretty sure Amos mentions having a brother in Hokkaido, when he’s talking to his lawyer on the phone or something.