I'm wondering whether I can remove textures from the "Texture Slots" panel in "Texture Paint Mode" because I can't see any button or menu for removing it.
It's like with unused Materials: don't use them, Save, close Blender, open Blender and they are gone.
In modern Blender you can go to File > Clean Up > Purge All to remove all unused Data Blocks.
Thanks, spikeyxxx! I'm saving my textures quite often with an incremented index but in the "UV Editor" it only shows the name of the "Color Texture" or the "Bump Texture" that are also displayed in "Texture Slots" panel in my "3D Viewport". Nevertheless, Blender always shows me the last texture versions when reopening it. This is a little bit confusing. I would expect the "UV Editor" to show me the file name of my last saved texture file. Can I understand the "Texture Slots" as a container with a label which internally links to the latest saved texture file (the one with the highest index)?
In modern Blender you can easily see what Texture Slots are; look in the Shader Editor:
The Slots refer to Image Texture Nodes and in above screenshot they are all empty; there are no images made yet, the Image Editor is blank. When deleting the Image Texture Nodes, the corresponding Slot might stay, but when closing and opening Blender they will be gone (I think this is called 'orphan data').