This came up with my kid when a chef show was on. After I tried some of the explanations mentioned upthread, he thought for a moment and said:
Any questions? 
Maybe it’s just me, but I’ve never encountered Americans using the word that way. The only ones I’ve heard are expat Brits living here. In fact, it happened just the other day. A bunch of us were in a restaurant, and the waitress (a Brit) said something about sweet vs savory, and we all went… huh? None of us knew what she meant.
Why do you hate America? 
I can totally see this.
What is the flavor chain created in turkey crisping, and in the roast beef browning?
That is a classic ‘savory, meaty’ flavor, and is indeed called browning, or some such.
That is a pretty amazing leap for a child, however… How old?
I think he was 11 at the time. And I just kinda stopped and said “well, yes, yes it does…”
I still have a problem wrapping my head around a taste that puts beef and spinach (even when cooked) in the same category. Don’t get me wrong, I like them both, but they don’t naturally fall in the same taste category for me. Maybe I was just conditioned to think of meat and vegetables as being different things.
Which reminds me… a vegetarian acquaintance once asked me to describe what a good steak tastes like, as opposed to a mediocre one. I had to resort to the portobello mushroom analogy for lack of anything else. Maybe I should have said: Really good spinach!
Savory has always meant “not sweet” to me.
This is what I am talking about, with the browning, and I think it is what you are looking for, John Mace, to describe the good steak flavor. It happens to beef when it is cooked right.
The Maillard reaction.
Savory is also the name of a particular spice.
This sounds about right to me, but I’d add
Spicy - caused by capsaicinoids
Probably not technicallly correct, but it seems like it is.
Just one wise-ass one:
You mean like a Hershey bar?
Yeah, as I understand it, spicy isn’t considered a flavor on its own because capsaicin doesn’t trigger taste receptors.
It certainly affects the taste of the food, but through a different mechanism. I’d put spiciness in same category as things like temperature, moisture, texture, and astringency.
Data point: the friend I heard this from is American, but she married a Canadian.
And there’s also different kinds of spiciness. The spiciness of mustard or horseradish is different than heat in peppercorns, which is different than the heat of chile peppers.
Pretty much “not sweet” for me.
Oatmeal with brown sugar=sweet.
Oatmeal with herbs and maybe an egg on top=savory.
Reading this thread made me note one thing. Hearing/reading the word “savory” makes my mouth water a bit (I am not a foodie by any stretch). So, for me, its a “taste” that makes the mouth water (I don’t get that for the other “flavors”). Exactly what that is…well thats what this thread is about I guess.
It just means “seasoned without being sweet” to me. As a vegetarian I definitely don’t take it to mean “meaty” as that would mean that nothing I eat is savory, which is a falsehood.
I grew up in the States, and always understood “savory” to mean “tasty” (I help with a Feeding The Homeless event called “Savory Sunday”).
But then I was greeted at an Aussie friend’s home on a Saturday morning with “I made scones! There’s a plate of blueberry ones here, or if you’d prefer savory ones, they’re in the kitch.” I thought “Are you saying only the ones in the kitchen taste good?”
So I embarked on a year-long research project, and decided that it does indeed involve “umami”. When I first heard that term, I remembered as a younger lad tasting something and thinking “Oh, it’s got that Dark Taste I like…”
Rule of thumb: If the average person might eat it as (or along with) the main course for lunch or supper, it’s probably savory. If it’s more likely to be dessert, then it’s sweet.
Sweet potatoes/candied yams - they’re a main dinner side dish, not dessert, but are sweet.
Yeah, in day-to-day American speech, you rarely hear “savory” and it’s not common in eating-out speech. There are people who use it, but they tend to be the ones who have some knowledge of food. The term is also used in advertising (“rich, savory goodness”) in which it just means “tastes good” without specifying any particular flavor.
“Meaty” does not mean “has meat in it.” It means “having some of the culinary qualities of meat” and there’s a lot of vegetarian food that can accurately be described as having a “meaty” quality, including almost any kind of mushroom.