You understand what the Federation is, don't you? It's important. It's a peacekeeping and humanitarian armada...
In other words, I hear where you're coming from. Would you feel better if I mentioned that countries do send civilian ambassadors to NATO?
The passage I quoted made me think of that exact line too!

In case there was any doubt, my comment certainly wasn't intended as any disagreement with you; I was just surprised to hear such a thing from David Gerrold. I just needed to wrap your quote around Gerrold's quote, otherwise it lost the link attribution to the previous post.
(And TBH I didn't know that about NATO. Although I thought it was more of an international organization, than a single military force?)
Yeah, NATO is not a military force itself. Rather, it's an international military alliance comprised of those sovereign states that have signed and ratified the North Atlantic Treaty (hence its full name, the North Atlantic Treaty Organization). Technically speaking, NATO Member States don't send
ambassadors -- they send what are called Permanent Representatives to the North Atlantic Council (the Organization's decision-making body); "ambassador to NATO" is an informal name.
NATO would be more akin to whatever joint Federation/Klingon/Romulan military command structure might have been set up during the Dominion War than to Starfleet per se.
I too thought of that line in Star Trek (2009), which drives me crazy. One, it confuses the Federation and Starfleet, and two, who doesn't know what country they live in! It's like me saying to a neighbor, "You understand what the United States is, don't you?" and then defining the Coast Guard.
Yep! Confusing the state with its armed forces is weird. I mean, I get that ST09 was designed to attract newbies and they needed a quick way to explain what the Federation and Starfleet were to people who'd never seen
Star Trek before or who were only casually familiar with it. But the line should definitely have been something like, "You understand what Starfleet is? It's a humanitarian and peacekeeping armada. The Federation unites dozens of worlds under one flag, but Starfleet is the glue that holds them together!" or some such.
* * *
Having said all that, we repeatedly hear of Federation members who have ambassadors to one-another or to the UFP itself, and that always seems weird to me. Like, the State of Ohio does not send an ambassador to the Commonwealth of Virginia; the Commonwealth of Massachusetts does not send an ambassador to the United States. In fairness, though, the
Länder (federated states) of Germany have what are called
Landesvertretungen in Berlin that support their state delegations to the Bundesrat (which is comprised of representatives of the state governments, like the U.S. Senate before Senators were directly elected), and the
Landesvertretungen are basically
de facto embassies for their state to the German federal government. Maybe the Federation has something like that -- maybe Federation Member States have something on Earth that's not
officially called an embassy but which serves the equivalent function and is unofficially called an "embassy?"
I'm still not clear on what the distinction is between a Federation Member State's Federation Councillor and a Member State Ambassador to the Federation. Was Sarek the Federation Councillor for the Confederacy of Vulcan, informally called "the Vulcan Ambassador to the UFP?" Or maybe some Federation Member States retain an office of "Ambassador to the Federation" that's mostly symbolic? Beats me.
The reference we hear of Federation Member States engaging in direct diplomatic relations with non-UFP sovereign states makes sense in the context of
paradiplomacy, though presumably the Federation government has final say on such relations.