Is there a site that give correct data for big chess database?

[Uri Blass]

If I search in lichess statistics about 1.Nf3 Nf6 2.c4 I get exactly the same result as I get if I search statistics about 1.c4 Nf6 2.Nf3

I would like to see different statistics because the population of games that started in one way is different than the population of games that started in the second way.

Maybe people who reply to 1.Nf3 with 1...Nf6 can get 1.Nf3 Nf6 2.c4 but never get the position against 1.c4 because they reply e5 so they know less theory so 1.Nf3 Nf6 2.c4 leads relatively to better results for white but unfortunately I cannot check it based on searching lichess database.

Is there a site that allow me to search based on exact moves with exact order of moves and not only based on position and also show statistics of win draw loss like lichess?

I assume that lichess simply memorize the number of times in the computer and I guess memorizing it is more memory in the computer for them and maybe they do not have enough memory but I think that it is possible to do it not for every line but at least for lines that appear at least X times for some X and memorize the result in the computer.

For lines that appear less than X times we can decide that statistic of specific order of moves can be misleading because of statistical noise and show the user estimate for expected result that does not consider the order of moves the position happened.

[jefk]

depends on what you think are 'correct' data (transpositions or not); nowadays most databases
seem to use transpositions (by default, so not even optional); also online i presume (theoretically,
i.e. from a scientific point of view, i think this is correct; but when looking at the human/
psychological point of view it -maybe- can make sense to ignore such transpositions).

So, although it's not a 'site' , in Arena you can switch off use of transpositions in the (.abk) book (eg. rightclick on
book window, then book management). You can use a large .abk book, or simply make one yourself from a pgn.

And indeed then, 1.c4 Nf6 2.Nf3 gives different -recommended- book moves
than 1.Nf3 Nf6 2.c4 , when switching off use of transpositions...

[Dann Corbit]

This is an alternative:
https://www.chessdb.cn/queryc_en/
It is position based, so I do not think it will give you what you want.
But I think the data is very good.
I think your main idea is correct, but the place where a worse line will diverge is not the endpoint but somewhere between the starting point and the endpoint.
If all the evaluations are the same along both paths, then the divergence is probably only psychological