I suspect that terms might be quite local, perhaps even varying from city to city. In London we have the "oyster card". You would say "Top-up your oyster".
So "Top-up" is an expression for adding money to a payment card in British English.
Top-up used to be common in the US, but a quick scan of the internet suggests that current usage in the USA is that you "Reload" the card.
"Add money to the card" is long-winded, but would be understood everywhere.
Using "charge" or "recharge" is probably understandable in context, but it clashes in meaning with the use for credit cards. Charge also means "to ask for payment". So you can say "The bus driver will charge you $2", or in the passive "You will be charged $2". Also, to use a credit card you charge the credit card company when you buy something (and pay back the company at a later date). So "charge" can mean "to pay money from the card" not "add money to the card".
With the widespread adoption of contactless credit and debit cards, and phones with Google/Apple Pay, the use of pre-payment cards is becoming rare.