Re: your first image with 0.254mm clearance: This is fine. The holes are small enough that they won't show up in practice - the solder mask process doesn't have that kind of resolution, at least at the "cheap PCB" level.
I should note that I run with 0.025mm clearance on my own exports - enough for the fabs to get the hint that they shouldn't go nuts (like when you have zero clearance), but they'll increase it to whatever they need for manufacturability reasons. I've never had registration issues where the mask ended up on the pads.