You need to match the center pin and the barrel.
Your PS's have a center pin of 2.1-mm and a barrel of 5.5-mm diameters for both. Your 2nd link (BJ-002BH) has a pin diameter of 2.5 mm. That will not fit into a 2.1-mm hole without a whole lot of effort.
Your first link has a pin of 2.0 mm. That will fit into a 2.1-mm hole nicely. The outer dimension is less critical, as the contact is spring loaded. BTW, 0.1 mm is slightly less than 0.004", or a loose slip fit.
Given the two options you present for a wall wart, go with the first PCB connector. As for the wall wart itself, you need to decide between a bridge rectifier circuit and a full-wave circuit. Such comparisons are aplenty on the Internet.
John