The claim reduction by 50% seems to be an oversight. Haven't had time to check if it's just a tooltip error.
As far as distant overseas, the patch notes are misstating what's actually going on. It's not a 50% reduction in coring cost for distant overseas. It's a 50% reduction in base tax for coring calculations.
For example, a normal 4BT province would normally cost 80ADM to core: 4BT * 20ADM/BT. With a claim, that would reduce by 25%: 80ADM - (80ADM*0.25); with claim + Adaptability, that would reduce by 50%: 80ADM-(80ADM*0.5), because reductions are additive. Additive bonuses are why it was theoretically possible to get negative coring cost. As an aside, this is why games like EVE Online use multiplicative bonuses for such things.
With the new patch, a distant overseas province's base tax is reduced by 50% in the coring cost calculation. For example, a 4BT distant overseas province becomes a 2BT province for coring purposes.
This is also a stealth nerf to the Administrative idea group for Europeans expanding in Africa and Asia. Its second idea, Adaptability, would ordinarily justify taking the entire idea group, because it would pay for itself after coring 80 base tax -- 400ADM/(5ADM/BT Cored); with this change, it takes 160 in distant overseas base tax for Adaptability to pay for itself. For comparison, the India region has about 450BT.