I think it would be confusing to have two different "search" actions doing different things depending on the current view, no?

NO

Sorry, caps lock got stuck, my machine crashed, and that was the only thing which went through even though I didn't tell it to send it. I will reply with a proper reply in a moment. That "NO" by itself looks a little bit rude, but I assure you that I definitely did not mean to reply like that.

Proper reply:

No, I have looked at the design and in the "All" section there is a search, in the "Installed" section there is no such search nor can one press CTRL + F to get any kind of search so I don't think that it would be confusing for it to have its own search too as the sections really do appear very different and I don't think one would confuse the searches.

But I guess that you could change the design of the search so that it's even clearer, for instance you could have it take you to its own search sort of page (though exiting the search unless done so through clicking an item would return you back to the section you came from) where it would have boxes ticked or unticked depending on the section you came from.
So you could have tick boxes labelled "Not installed" and "Installed".

It isn't clear to me why this change is needed. Search results narrow down pretty quickly in my experience, so that filtering is unnecessary.