In that case every bank, insurance company, etc in the world is ”fintech” and the term is meaningless

True to a degree, but most would say there is a difference between standard bank "x" and a company that has billions in trades going through it. A fine lined distinction though.

Most banks don't have lots of people writing complex algorithms for their day to day job.

Most banks don't have complex algorithms (the sort that some would consider OCaml or Haskell a good fit for) in general, they only have complex software because they have complex business and regulatory requirements, complex and incoherent existing systems to work with, cost constraints and organizational issues.

I'd agree with that. I don't work in banking, but definitely understand how various regulations can complicate things.

I agree. I was in technology for in-house use in both commercial and investment banks in the 90s. I definitely refer to my work back then as fintech as much as my more recent work on algo trading systems for direct client use.