No, debit cards in most of the world do not function for remote transactions, even if they have a Mastercard or visa logo. My bank's cards for instance are Mastercard's "Maestro" product which is in-person and debit only, and has alternative infrastructure for online payments.
Which parts of the world are these? I’ve used debit cards issued by various European and US banks for online payments all over the world since the ‘90s.
What part of the world is "most of" it in this case? Where I live, Maestro and the like are considered children's cards for the reasons you mention, and real debit cards work just fine for online transactions.
