What is a bank?

Via Matt Yglesias, Justin Fox writes:

If you borrow short and lend long, you’re effectively a bank. It’s
becoming ever less clear to me what justification there is for nonbank
borrow-short-lend-long-institutions other than regulatory arbitrage.

I cannot sign on to this principle.  Business-to-business trade credit is huge in the United States (it’s not all short term) as is commercial lending, as you might get from General Motors or for that matter Sears or Nordstrom.  Pawn shops are more common than you might think.  If we regulate these lenders like banks we presumably have to give them comparable privileges, which could mean discount window access, FDIC access, and the use of Fed Wire.  Plus they would be subject to other regulations on banks, including restrictions on affiliations with commercial firms.  I don’t want to do that and in many cases I don’t even see what it would mean to do that (how can we stop Sears from affiliating with a commercial firm?).  I conclude that we cannot escape some important legal distinction between bank and non-bank lenders.  Admittedly any such distinctions can become more problematic with the march of technology and the increasing sophistication of regulatory arbitrage.