I don't see anywhere option to disable the Search Activity. When you use for example Google Analytics you don't need it. How disable it?

In my opinion should be a built-in option.

I agree. Previously in IP.SEO as independent application it was possible by disabling a hook. I don't see a way to disable it in 3.4.3, but in IP.Board 3.4 Dev Update comments an option is mentioned. Am I missing something?

I do not want to fill my database with this kind of records and I would like to avoid unnecessary SQL errors like "Data too long for column 'keywords'..."

Those comments in the Blog about being able to turn off the search activity are incorrect, it can only be disabled by doing an edit to the source files. There's no setting that controls whether it logs or not, and since it's done in the registry you can't even write a hook that would stop it.

+1.

When we have powerful tools such as Yandex.Metrika or at least Google Analytics, then Search Activity becomes only unnecessary throwback.