Okay, now I understand, but grandparent was talking about who was in the lead, not who was first. You generally can’t stake claims retroactively, so USSR could have made a claim, but failed to. On the other hand, it’s tough to distinguish regions of space, so I’m not sure it could stick.

They will care about who can make a defensible claim and defend it, on whatever real or made-up grounds. That will be 99% earthly politics and power projection and 1% space-to-space lasers for a long time to come.