Zeyzu wrote: When I create a new photo album, why do the photos I add there still appear in all the photos?
Because it's " all the Photos."
You didn't tell us, but it sounds like you're talking about an iPhone rather than a Mac, is that right? You also didn't answer the questions about what OS you're using. Without the information we asked for, it's harder, but I'll try to make this fit.
Pictures aren't exactly "in" an album. When you "put a picture into an album," its name is added to a list of pictures to display together when the album is clicked, kind of like a music playlist, but for pictures. So two albums can both have the same picture name in their lists (like two playlists with the same song,) and that picture will show up when you click either album, but there's only one picture file-- it's just on multiple lists. And when you remove a picture from an album, its name remains in the lists of other albums, and the file still remains in your Library.
So albums give a specific view of your pictures. The picture of "Aunt Ethel at the Grand Canyon" can be in the "Aunt Ethel" album with other pictures of Aunt Ethel, and it can also be in the "Grand Canyon" album with pictures of other people at the Grand Canyon. And it can be in a the "September 2015" album. Each album is pointing to a single file stored in the Photos Library, so having pictures in multiple albums takes up no more storage space. As you can imagine, this is very powerful in organizing pictures.
The Library View isn't really an "album"-- it provides a view of all of the pictures in the entire Library in the order of the date they were taken. You can't remove a picture from the Library View without deleting it entirely. And Recents is a view provided by Apple to show all of the pictures in your Photos Library in order of the date they were added. This can't be changed. These special "albums" always provide the same view so that, when something goes wrong or seems off, you (and we) can depend on them showing the same thing, and there's no chance that someone snuck in and changed the order or removed pictures.