Eddie is right. "Caminando" is all right and correct. It isn't Latin American usage because we also use it here in Spain.

As for "paseando" there is a sligh difference. "Paseando" implies you are "caminando" but mainly for pleasure. That is, you are watching the windows shops, enjoying the sun... When you go to work everyday you don't go paseando. You go "caminando" or "andando". Both are correct.

And about the "Camino de Santiago" if you do it on a bycicle then you are not "caminando". You are doing it "en bici".

Saludos,

Antonio