In the new series, at the beginning of Casino Royale Bond literally 'earns' his 00 by killing 2 people.

He's not a spy, he's a thug, and the movies aren't really confused about that.

Neither are the books. James Bond is a killer who found a respectable job.

CIA, KGB, Mossad, etc. have long and colorful histories of doing much more than writing reports.

The contemporary CIA has its own drone assassination program in-house. They’re not all flown by the Air Force.

I think it’s fair to call a member of an intelligence agency’s foreign service a “spy” even if their job is more covert action than intelligence gathering.

This could depend on the motivations of their contact. If they need to impress e.g. warlords in order to manipulate, there is a good chance they will arrive armed. This is after all their ticket to the access they need, and they may only have one chance. Access operations are tailored to fit the target.

Wet ops.

https://en.wikipedia.org/wiki/Wetwork

Would that mean that any assassination done with a weapon is a failure?

I would think the primary job of a spy is to be a manager (a handler) of a bunch of traitors (including unwitting ones) in the enemy country. Any eventual assassinations would be best done through the hands of one of those subordinates.

This is more like the picture le carre paints

No, it would mean that spies don't do assassinations.

The primary job of a spy is to gather classified information, ideally without anyone noticing. Dead bodies tend to be noticed.

Actually, in all the Bond movies, I don't recall him ever having a mission to spy per your definition. Spies are typically used in situations between countries and many of Bonds foes clearly weren't countries, e.g., Goldfinger, Dr. No, Drax. They were just super criminals without allegiance to any country. The closest classic spy movie I recall was From Russia With Love but even that involved SPECTRE more than Russian.

I mean, fundamentally the spy thriller spy does anything a given country needs done in another country that would otherwise be considered illegal or an act of war, if the provenance of the order was known.

Bond was more of an assassin AND spy. A spassassin.