For example, one of the order's memebers confessed new members were made to defile the bible and spit on the cross during their inauguration. This was supposed to prepare them against Muslim torture, he said. This explanation sounds highly dubious to me.
As all confessions came out of torture and was then later denied, those confessions are of questionalbe quality.

About them who fled to Scotland, they merged with the Hospitaliers there, the Hospitaliers got even a name change to the Templars of St John or something like that.

I think the huge crosses in sails used by Portuguese (cross of Christ) and Spanish (Cross of Santiago) ships in the beginning of the discoveries could be a sign of their sponsor or a sign to be recognized by fellow christians in distant lands.
It worth noticing that the reconquista was still more of less active at the time. Grenada fell 1492.