The assumption doing the most work here is that "same-day" has a self-evident meaning in the term sheet. It almost certainly does not, and that ambiguity is the actual problem.
Three different clocks are in play: the funder's wire desk, the county recorder's timestamp, and the calendar date on the deed. None of those are the same thing, and the term sheet apparently does not specify which one controls. That gap is what this dispute is sitting in.
The funder's position, that their wire desk closes at 5 p.m. and anything after that rolls to next business day, is an internal operational rule. That rule is enforceable if it is in the term sheet. If the term sheet says "same calendar day" or "same recording day" without referencing the wire desk, the funder is trying to substitute their operations manual for the contract language, which is a different matter entirely. The county log showing same-day recording is the strongest external evidence you have that both legs closed on the same calendar day.
The risk you did not raise: this funder's interpretation, if it stands, means any B-to-C closing that runs past their wire cutoff becomes an overnight deal regardless of what the deed or the county says. On a 2 p.m. scheduled closing with any title company running late, that is almost every afternoon deal. That is a structural exposure on future transactions, not just a $600 dispute on this one.
Whether the term sheet language creates an enforceable right to the same-day rate is a contract question. A real estate attorney in your state needs to read the actual term sheet and tell you that. I won't speculate on the outcome.
What does the term sheet say verbatim about how "same-day" is defined? The exact language is what matters, and it would change the analysis considerably.