Bank to bank = interbank. Intermediaries who don't charge a fee do not affect the outcome. The money I spend at Kroger goes from my bank to Kroger's bank, I'm pretty sure (and am also willing to be proven wrong), not to the Kroger store itself for them to then put into their bank. I don't know what the legalities are in terms of defining who's what, but the practical result is the same.
Apart from the debit card system, though, most banks do offer their own bill paying services where you simply "write a check" from your online banking account to whoever you're paying, and then it's a straight EFT, the money goes from your account to theirs. The recipient has to provide you with some kind of ID, if not their actual account #, then some other kind of code so the computer knows where to move the money.
We as a country might well be naive about how fluidly money moves electronically these days, but boy, does it ever!