It is 100% correct to refer to a bank as a lender. This is standard use of terminology.
If you check the Promissory Notes, I'd guess they probably don't say "lender", except as possibly an alias for a definition that is not equivalent to the standard definition -- wherein, one forgoes actual physical property (cash, for example) for another's use; with the expectation that the property will be returned at some point (possibly with some form of payment to compensate for the real loss of property).
Edit: Do you think Wells Fargo has myriad $100k bundles just sitting around waiting for students to take? I don't think so. They use their power to create new money, rather than loan out the entire principal from their reserves.