Interesting
After how many moves can you tell person's strength?

It varies. I'm no master by any means and often blunder in less than 60 min games. But in the past week I completed a daily game that chess.com said had 36 excellent moves and 4 good ones and the CAPS %age was equiv. to around 2600, and my next game was a Rapid 15/10 game with 15 excellent, 4 good, and one (minor) inaccuracy with a CAPS equiv. of around 2200. Then in the next game I blundered.
The first two opponents would have considered me a very strong player, the last one would figure I knew a little about chess but didn't have it together.

It depends on whether you're playing a well-known line or a more obscure one. At some point your opponent has to be making up his own moves, rather than just playing them from memory. Of course, if he is playing from memory, he's probably a decent enough player.
You know they're a strong player by how it feels to play against them. If they're moving at lightning speed and spitting out one good move after another, then you know.

Well, sometimes it takes more than 1 game, but yeah, the first few moves out of book give you a general idea.
Sometimes the fact that they know book... I had a guy tell me he was rated 1000 and we sit down to play some blitz, and less than 5 seconds into the game we're past move 10 in some mainline... so obviously I already know he's not 1000.
(He wasn't lying, he just hadn't been to a tournament since he was a little kid.)

Once you get to the middlegame, you can really start to tell the strength of the player.
If you see weird moves they can be a really good player too... or a really bad one... If they are not that good of a player you can usually tell if they play the philidor defense ;-) or they play poor opening moves.

Once I was paired with a 1600 guy at a tournament, but in the crowd I incorrectly read his rating as someone else's, so I thought he was just under 1900.
During the game, after the first tactical mistake, I thought oh, that's too bad, he must be having a bad day and miscalculated. After the 2nd big mistake (this time positional) I wondered if maybe he was overrated. After the 3rd mistake I knew something was wrong. Either the guy was sick, or they'd misprinted his rating or something.
I have similar experiences playing anonymous online blitz. After the first mistake I'm often thinking to myself "ok, either they suck or they're really good and just messing with me."
All this to say, it's not an exact rating, but there's a range, and over time you narrow it down.

Every player has two levels, or strenghts: one is the average strenght or level for a certain amount of games and/or time. The second, is how you are atm. Of course, how you are atm can never be too far from your average level, but still: you can alternate very good games with very poor ones, and, in that aspect, you might be able to guess the "level of the moment" of a player out of a few significant moves, but never their average level, which is rightfully (most of the time) reflected by their Elo.
Or how would you define the strenght of a player scoring =2215 -1520 =2235 within the same tournament and in a row?

it depends , could be move 2 could be move 52 , but i get a "hey this one knows what they are doing"
then i know

It really depends. I was once playing a casual game with black that went 1.e4 c5 2.c3 and I was like, ''he knows the Alapin, so he means business,'' and I think I did 2...d5, he let me take on e4, and then he played Qa4+ to take back the pawn with the queen or something. So then I thought, ''Ok my bad.'' Usually, I look at it like this: Playing with the queen, not knowing any opening, blundering, weakening pawn moves. If nothing like that happens I know it's at least a decent player. So, I can have my suspicions in the first few moves, but it can take up to the (late) middle game to be sure. Some positions are not that sharp, but if they're much lower rated they will slip up some time. Some players are also very defensive, so then it takes a full out assault before the cracks begin to show. That doesn't mean they are better than others, the game will just be longer.

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