The exceptions in the Washington/NYC case are small countries for whom it makes economic sense to only run one diplomatic office, and who have chosen to run that office in NYC and accredit it as the embassy as well as their diplomatic mission to the UN.
(Also a side note - between Commonwealth countries the term used is High Commission, not Embassy - but it's the same function).
kind of weird.
For example did this one get counted?
Polish Embassy in Roma, Vatican
Embassy of Poland in Vatican City
Via dei Delfini 16 int. 3 00186
On the other hand, many country have one single embassy for Italy, Vatican and Sovereing Military Order of Malta. Plus, some also incorporate the Embassy to the UN in Rome.
On Wikipedia - indicated as source - it says: "the capital city of Rome hosts 141 embassies for Italy as well as some for the Holy See of Vatican City" which by 1 is in any case inconsistent with the number here (140)
It should have counted.