It's possible you & I are simply looking at paladins in too different a way to agree, but I'd say that definition certainly proves paladins cannot be secular. As holy warriors, they are absolutely connected to the religious, spiritual or sacred. They draw their powers from one or more divine sources, and you can't get much more sacred than deities :)
The thing is, Paladins are not described as Holy Warrior in the PHB. If they were, it would be a different story.
Those are actually eight different definitions for the use of that word, not a single definition. The one that applies is layperson.
I am not saying Paladins cannot be ordained, but I am saying they don't have to be.
It is a joyful thing indeed to hold intimate converse with a man after one’s own heart, chatting without reserve about things of interest or the fleeting topics of the world; but such, alas, are few and far between.
– Yoshida Kenko (1283-1350), Tsurezure-Gusa (1340)