Banks change "the fine print" practically with every breath.

But, give us some clarification, please.
So you opened a "checking" account at the bank.
Now you're trying to open a "PERSONAL checking account"??

What's the difference and do Spanish banks even have "checking accounts"?

Do you mean, possibly, instead of a "checking account", a "debit account" with a debit (credit) card to make electronic purchases?

Since you already have the Caja Madrid account, you can't open an additional service therein? Or, I gather, this also has something to do with your mobile phone contract payments, correct?

Saludos, MadridMan