I think the default image view in Photos is photos taken by the phone, arranged by date (I say "I think" because I don't use the app much). "Library" mixes 2 different types of collections of images: "Albums" are collections of images in Google Photos, which are in the cloud, while "Photos on device" is actually the list of different folders in your device's storage that contain images, and those are on the phone but not in the cloud unless you have told Photos to back them up. So it sounds like you are seeing the contents of a "Screenshots" folder on your phone there. If you use a file explorer to look around your storage you should be able to locate these folders, which might give a clearer idea of where things are really stored.
Avoiding saving to the cloud by default is easy: turn off the backup in Google Photos (since on a Pixel that's the obvious thing that would save to the cloud).
To allow photos but not screenshots I'd look in the app's settings and see whether there are options to choose which folders are backed-up that way. Photos and screenshots will be in different folders, so if you can choose which are backed-up/synced that will do what you want. I can't tell whether there is any control over this because I have backup off, and so the only setting I can see is one to turn it on - there may be other options available if I do, but I don't want to do that in order to find out.
