You’re welcome. On US terminals, ALWAYS select ‘credit’ - credit means to run it over the major global network (e.g. Visa/MasterCard/Discover/Amex/etc) that the BIN corresponds to.
Debit, on a US terminal, means to run it over a patchwork of US regional debit networks. By law in the US, all debit cards from major banks must support at least two non-affiliated networks. Generally, they support Visa or Mastercard (rarely Discover) - aka the ‘credit’ networks, plus a handful of the regional debit networks.
With a foreign card, always select credit, unless it is a Canadian Interac card, in which case select debit (it can run over the US regional NYCE network, and Interac isn’t a global network).
Clear as mud?
Also, you should only ever get the credit/debit prompt on a magstripe terminal - EMV (chip) terminals will see the Monzo card only supports Mastercard and that’ll be that. Application selection works completely differently from magstripe routing.
tl;dr: ‘Debit’ on a US terminal is only applicable to US (which can also work if you pick credit) and some Canadian debit cards. Always select credit with the Monzo card. If prompted for a ‘ZIP’ code with Mastercard, which is a US postal code, enter the numerical digits from your postcode, then add zeroes to hit five digits. With Visa, I believe you start with zeroes, again the goal is five digits. Amex - I have no idea. I’ve never found anything that works.
Hope this helps some!
P.S. if you want some further fun confusion, in the US ‘debit’ and ‘PIN network’ are often synonymous and have no relation to how those terms are used here. The US regional debit networks mostly require online PIN (as you saw) for both magstripe and chip transactions. Visa and Mastercard (the ‘credit’ networks) are signature for magstripe and ‘chip and choice’ (they don’t mandate what CVM banks use in the US) for chip, but almost all American banks program their Visa and Mastercard applications to prefer signature. Thus if you’re ever asked by a cashier swiping your card if you want to enter the PIN or sign, always say sign - that’ll have them pick ‘credit’ 
BUT, if they’re inserting your card for a chip transaction (except for ATMs and fuel pumps which are insert for magstripe, most terminals in the US insert for chip and side-swipe for magstripe, just like here), make sure you say ‘PIN’ because otherwise they’ll just hit enter and attempt to bypass the PIN, which is likely to result in a decline (not all UK banks will decline this, but many will, and I’m pretty sure Monzo does from posts on here).