Learn English – “Cancel” versus “close” versus “abolish” when referring to a debit/credit card

A debit card is a card issued by a bank, therefore it qualifies as a bank card.

A bank card is not necessarily a debit card, because it could be a credit card etc., as explained in one of the answers to the question you linked to.

debit card -> bank card

bank card -> debit card, credit card, etc.

Because of the relationship, I would say that a debit card is a type of bank card, but not a synonym. The best way to check if bank card works as a synonym of debit card is a simple substitution. If someone said:

Please give me your bank card

Someone who had both a debit or credit card in their pocket might hand the person a credit card, instead of a debit card. Sometimes, you would know based on context which card they wanted you to give them, but it is not explicit. It is however, very obvious which card the person wants if they say:

PLease give me your debit card

To add to what others have said about the connotations -

I would say that:

• Utility sometimes speaks more to how something is used and what makes it useful. The utility of a swiss-army knife can refer to what you can do with it, and perhaps even how.

• Usefulness sometimes speaks more to whether and how much something is useful. In the wilderness, the usefulness of a swiss-army knife is generally greater than that of a clipboard.