Well, what I emphasized in the excerpt above is that the E-Bill system can be used anywhere in the world. So I think you'd be able to use that just fine.
As for a "check":
Checks are good across institutions. It doesn't all have to be the same one.
For example, say that my bank is called Bank of America. I open an account with them. When I do that, I put some money in the bank, and I get a booklet of blank checks. They look something like this (but not filled out):
If I want to make a payment -- to anyone who accepts checks, which is a very large percentage of all merchants, and across many, many different institutions -- I go ahead and write out a certain amount on my check. That amount must be less than the amount in the bank, because when the recipient deposits the check in their OWN bank account, the money will be removed from my account and put into the recipient's bank account. (Those banks may be different. The recipient doesn't also have to be a member of Bank of America to get the money.)
So for example, if you have $1,000 in your account to start with, and I also have $1,000, and I write you a check for $500, after you deposit your check I will have $500 left in my account and you will have $1,500 in your account.
Does that make sense?