Textured view will display the image texture applied to the color value.
My red material looks like this, and the white material is identical, with a different colour.
In textured view, they’re both white, as no textures are assigned. I dont know why yours display as red, and i have to look into the blend file to draw a conclusion, as texturing is not my strongest side.
However, if i add a red texture like this:
i get this result in textured view:
After unwrapping it, i get this:
Now that i do have a texture applied to the model, i can locate it in the UV editor, and edit it there as i please:
Now, in texture paint mode, i can draw freely as much as i please. I cannot paint on the white part, because no texture has been assigned, only a material.
A texture, and a material are very different. A material can use textures, but a material on its own is not a texture.