That is a confusing explanation.
First off, understand that I don't think in metric, so bear with me if there are any conversion errors.
One thing I do is specify in my fab drawing notes is that the listed sizes in the drill table are the finished hole sizes. I don't worry about how they (the fab shop) get there.
In fabrication, the actual hole size drilled for a finished plated hole is a bit larger than what you specify. The oversize I'm used to seeing is usually 3 mils (or ~0.076mm) +/-3 mils (or ~0.076mm) for plated holes, and 0.0 +/-2 mils (or ~0.05mm) for non-plated holes. The finished hole size obviously carries the same +/- tolerance (2 or 3 mils) as the drill. So your pad sizes also need to take that plus any layer-to-layer offset tolerances (ask your fabricator what tolerance they meet) into account.
But your shop oversizes their drills for plated holes quite a bit more than what I'm used to seeing, 4 mils (0.1mm) for 18-mil (0.45mm) and down plated finish holes, 6 mils (0.15mm) for holes larger than the 18-mil (0.45mm) plated finish. And they do list a 2-mil (.05mm) oversize for the NP holes, which meets the tolerance I'm used to seeing.
The minimum annluar ring is the minimum amount they require the pad to be over the hole size on any one edge, their MAR is apparently 0.125mm
Anyway, what they are saying is for a finished plated hole size of 0.3mm size you specified:
Add 0.1mm (because it's smaller than 0.45mm) = 0.4mm
Then add the MAR: (0.125mm x 2 = 0.25mm)
Then add the numbers (0.25mm + 0.4mm)
So, the minimum pad diameter for a 0.3mm (~12-mil) hole is, per their specs, 0.65mm (~26 mils). which sounds about right. I'm guessing this is a via?
They don't say what size pad you're using, but I'm guessing it is smaller than 0.65mm?
BTW, I don't know about that shop, but you can often waive requirements, as long as continuity is not compromised.