While they are functionally meaningless (I completely agree with you in that regard: 20 HP out of 200HP is the same exact thing as 200,000 HP out of 2,000,000 HP), somehow big numbers don't feel meaningless, and that's why I tend to be a proponent of medium to big numbers, more than most designers are.
They are also practically and psychologically meaningless to a video game player as well. Video game players have their own culture, as you describe with the Japanese. So, while dealing with regular Western Culture, you would be correct...
We have to make a distinction between Western Culture and what I'd call "Player Culture".
Players become "desensitized" to things after a few hours of gameplay. Your beautiful graphics you worked years on to make as realistic as possible? Players don't even notice them anymore after 3 hours of gameplay. It becomes "background noise" to the gameplay.
This same thing happens to Numbers. You even illustrate this point yourself with:
"I know that when one of my characters finally lands a hit that crosses the 1,000 damage mark (and to a lesser extent the 10,000 damage mark) and I get to see that 4 (or 5) digit number, it feels really, really good."
This statement here assumes you're doing damage lower than 1,000. Probably significantly so. So, when you finally do hit 1,000 damage, you feel pretty good. However, that's the context of "useless". If you started out the game doing 1,000 damage a hit, you'd feel "less good" about doing that much damage or even about doing 10,000 damage.
The human brain tends to "look for patterns" as well as "tune out noise". If you've been doing battle for a while in an RPG and land 443, 478, 503, 489, 462, 455, 491, 437, 324 hits... Which ones immediately stand out to you? Even in just this series, your brain immediately picks out the highest number (503), and the lowest (324). It does this through pattern recognition. Now, as a video game player, if you're used to seeing 440-480 damage on a hit, you're probably going to tune out numbers between there or close to there. That's because the game itself provides no functional difference between 440 and 480 damage. What players tend to gravitate towards is "killing monsters in one hit". They don't tend to worry about what the numbers say, as long as the enemies are dead in a single hit. The numbers can matter more against a "boss monster", but the concern is the reduction in amount of hits it takes to defeat the boss.
"Player Culture" tends to demand this sort of context. You can even see it prominently in MMOs and the way those players interact with the game. They avoid as much combat as possible (probably because it's better to not waste time than to see large numbers, which leans heavily towards players caring how quickly they can kill something rather than how big the numbers are while they kill it), they really only care about "DPS" damage against Raid Bosses (high level content where the game rewards as big of numbers as you can possibly get as players are expected to land hundreds of hits, if not thousands, before it dies), and players who can't output a lot of DPS because of their class restrictions (like healers) worry about their numbers being too large in context of their characters (there is such a thing as "overhealing", which is effectively wasting HP restored for MP spent if you heal too soon).
The game itself teaches the players the "context" for the size of numbers and what they mean. The problem is that in games... the size of the numbers doesn't matter. Except in relation to each other. While $10,000,000 is impressive to have for a US Citizen... Having 10,000,000 Gold, Zenny, Pokeyen, Gil, Col, etc isn't all that impressive... in fact, that's pretty common even by midgame in most RPG's.
In games, Numbers are effectively meaningless. They are disconnected from the culture of reality because they exist within their own culture.
A player really only cares how many hits something dies in. Namely, if it dies in one hit. If it doesn't die in one hit, why not? Is that number significantly lower in context to what that number has been for the 20 fights? Or, has that number suddenly spiked above "normal variance" for a boss fight? Is that a critical hit? Did it just shave off one hit I needed to make against a boss?
It actually works this way across all numbers in an RPG or a video game. Unless the game itself rewards small increments in those numbers going up, the player isn't going to care about those small increments. Likewise, the size of those rewards for numbers going up has an effect on how much the player cares about it as well.
There is a much easier way to explain this as well.
People don't value things very highly that they have in abundance. Especially if they didn't have to work very hard to get them (or at all). $10,000,000 to someone in the US is significant to most of the population because most people won't ever earn that much money in their entire lifetimes, even if they work until the day they die. It's an almost unheard-of amount. But, in a video game, that large number is very easy to attain within a few days, or a month at most. Maybe even in some extreme circumstances (and MMO's) that number is attainable within a year or two. But, the number is valued far less in a video game because of how easy it is to attain and how abundant the resource is.
1,000 damage means very little unless you had to work hard to get there, go through a significant amount of the game to attain it, or couldn't reach that number through any normal means. 1,000 means less if your Level 1 squishy healer using a dagger does that much damage in the first fight of the game.
Numbers are effectively meaningless because in a video game, they require context that only the game provides. So, 1 damage can be equal to 1,000,000 damage across video games. 5 damage can be equal to 50,000,000,000,000,000 across video games. The size of the number doesn't matter for anything except your damage formulas. It doesn't even matter for most players.
This is very true. Those last few numbers on a seven-digit stat are just completely worthless and probably won't feel that good to increase - and when there are too many insignificant digits tacked on (that aren't zeroes or abbreviations), it becomes harder and harder for the mind to parse what it needs to notice. (This is why I'm a proponent of medium-to-large numbers rather than extremely large ones!)
As a designer I tend to think the first three digits on any given amount (whether it's a three-digit number or an eight-digit number) are significant and worth considering, although I have a feeling that a lot of players only get excited about the first two.
Most players do only care about the first two digits as most RPG's only care about those first two digits. You aren't likely to have to take fewer hits to kill a boss or an enemy by having that third digit go up or down. Games just aren't designed that tightly controlled where 455 damage results in one less hit to kill than 454 damage... and 456 damage is also one less hit to the enemy you'll have to make. The larger the numbers you have in terms of damage, the less that third digit and every digit afterwards even matters. Those numbers only tend to matter in "slogs" of fights where you're meant to land so many hits that those numbers actually add up to something significant and DO shave hits off the enemy.
This is why I tend to prefer smaller numbers. I have no intention of trying to design a game in which battle lasts so long that I need a third digit... much less a fourth... and then have to make those numbers actually matter, rather than be a glorified "fraction" (when 432 damage would be no different than having 43.2 damage, it's a glorified fraction. It exists for the sake of itself).
