[LIV2]

I'm assuming that their bank cards and debit cards are two different things

Presumably their "bank cards" only use some local payment scheme rather than visa or MasterCard (explaining why it can't be used online)

The newer "debit card" uses Visa or MasterCard which means it is accepted anywhere Visa/MasterCard is

[t0astbread]

I'm Austrian and as far as I understood it's just a name change. Functionally it always was a debit card.

[LIV2]

Looks like currently they're using Maestro and switching to MasterCard Debit which uses the credit card network

Source: https://en.m.wikipedia.org/wiki/Maestro_(debit_card)