when my opponent is stronger than me, I play my best chess

same here. I lose to lower rated players a lot, but beat a lot of higher rated players

i think its the feel that if you beat this guy, you have accomplished something, so you play more carefully

Some people rise to the level of the completion. I'm kind of like that too sometimes. When I played at a local OTB club for the first few years the higher rated players they could not believe that I was only about 1500....but the lower rated players believed it. Eventually I got better by playing the higher rated players.

Also I noticed that the lower rated players just play weird moves....I waste time trying to figure them out.

I seem to play in a way that I can win or I can lose against any player rated 1200 to 2000, and in the same day. I think it comes down to strengths and weaknesses, and also matchups. Sometimes I can put a world of hurt on a much higher rated player and sometimes a much lower rated player will put one on me. It's pretty awesome when you beat a stronger player though for sure. 

Yes, good players play well in every game. And you are right on some other things as well.

On other things you may not be that right. For example: no, chess isn't about confirming own intelligence. I would say it this fits for only minority of players. The more you play chess and the better you become the more you realise chess and winning on chess isn't much about intelligence. If I ask someone who haven't played much chess he or she is most likely to say chess is related to intelligence. It may the reason it feels painful for them to lose or play material-odds.

I think most players think the challenge is the thing that motivates playing. For me that is true and I also think that chess game seems like a painting if the opponent knows as much as what is he doing. If he doesn't, I partly feel that he will ruin the painting sooner or later

You are right, that there have to be a lot of practise to win even lower rated players. But still there have to be the DESIRE to win and I think it's also something you can partly decide before the tourney starts and before the one game starts. I remember Anand had mentioned that every game have to be treated as a single individual game that need to be focused on. You have to forget ratings, money prizes and so on...I think it's possible to motivate yourself to focus on the moves and increase the desire to win before the game starts. The desire could be something like that you decide before the game that you will sit 5 hours on the chair and try your best on every situation, you will not try too easy ways (trading, playing like a wimp and stuff...)for victory when having an advantage, but mostly play objectively best moves (unless you have something like Q+R vs R, then you can sac the Q for the R-this is oversimplified example of course) and not get panicked when having down on material, but trying to create problems..

Maybe subjectively art for us is something that we are passionately with and we understand it enough well. Typically for person who has played 0 or few chess games in his whole life chess game doesn't seem like art, but just silly wooden pieces moving and he thinks it's mostly related to math and on some sort of (mathematical) calculation. Calculation is of course included, but you probably know what I am talking about.

Writing english seems bit difficult now and that's why my sentences are bit strange...

We understand you perfectly well 

Although to say you don't really care about the game against a lower rated player seems extreme, I find this problem in general very relatable.

I know in OTB play I feel very comfortable against players 300 or even 400 points higher than me, and often manage to draw or even win against them. It makes me feel like my rating needs to "catch up", but then I lose to players 300 points lower and it never does. I even once overheard an opponent suggest to someone that I was sandbagging my rating, which of course was not true, but further drove home the idea that my rating somehow needed to "catch up" to my strength.

I think a few factors are at work here - yes, psychologically, it is probably easier to play our best chess against a stronger opponent whereas when we feel overconfident, our playing may get lax and lead to mistakes. But also for us sub-2000 level players, blunders and tactical miscalculations just mean that we are both more likely to both win and lose against a wider spread of opponents. Moreover, when we win or play very well against a much stronger rated opponent, or when we lose to someone much lower, it really sticks in the memory, whereas when things go more or less "as expected" it is easier to brush off. Over time the memorable instances stack up and can lead to an incorrect assessment of things.

All this being said, I think the ability to convert "won positions" into actual wins rather than letting the points and half points slip by is closely related, and is something that really separates strong players from the masses.

So there are a bunch of my thoughts on the matter ¯\_(ツ)_/¯