What is it called when you “refill” a debit card?

Your question reveals a mindset about debit cards that simply doesn't exist in the US. Here, bank accounts came first, many many decades before anyone dreamed of electronic transactions. Thus, people have debit cards because they have bank accounts, not the other way around. You don't go out and get a debit card; you go out and open an account at a bank. This also means that you simply don't talk about refilling a debit card, because it was never filled in the first place.

You can say something like "I need to put money in my account" or "I need to go to the bank to make a deposit", but at no point would you mention your debit card, any more than you would mention your car keys when you're planning to drive somewhere.

There are some debit-card-like things that do have an upper limit and can be "refilled" — store gift cards, whatever they're calling food stamps nowadays, MetroCards, that sort of thing — but they're not called "debit cards", at least not without qualification.

If this a debit card that is linked to an existing account, you might say you were "adding funds to my account" or "making a deposit."

If this is a Stored Value card (perhaps bought from the store or used to disburse wages), then several providers say add funds to my card (e.g., Walmart). At Cardrex you may reload your card.

Square 1 calls itself a "Stored Value Debit Card provider." (These are neither mass transit cards nor store gift cards. They bear a MasterCard or Visa logo and I believe that this may be what the OP was aiming for.) It asks in its FAQ, "How Do I Put Money On My Square 1 Stored Value Card?"

In its answers, here are some other synonyms that Square 1 uses:

• Put Money on
• Loading funds
• Adding funds
• Have money placed on.

EDIT 1: Another term: reload

EDIT 2: Adding that Square 1 calls its cards "Stored Value Debit Cards" and listing Square 1 synonyms.

You have used the correct term from the user point of view. As a card-user you are expected to speak plain English.

I refill my card.
I recharge my card.
I top-up my card.

The correct 'technical' term will be some thing related to "crediting your card account".

I have a debit card and the bank's web site refers to this as "depositing to the debit card". I also work with a company that sells gift cards, which are essentially store-specific debit cards, and they refer to this as "incrementing the card" internally and as "adding to the card" when talking to customers.

I have a pre-paid cell phone and the phone company talks about "refilling the account".

It is also common to talk about "crediting an account", though this is perhaps a bit more generic.

In short, I don't think there's a single, consistently-used phrase in English. All of the above are used and understood.

In Canada, it is common to reload your debit credit card or reload your Starbucks card, or reload your prepaid phone card.

Reload, in the same meaning you would reload a gun, reload a stapler, reload a mechanical pencil, etc. Reload, as in "to add to".