.minecraft is just the folder where your Minecraft setup is, so all worlds, Minecraft's required files to exist/get it working and so on.
For multiple versions, yes the link above will make it clear but you will need to have different profiles through the Launcher to get it set up.
But in short you use the Launcher from the 'Launch Options' tab, create versions of the game that way, specific the game directory/file location of stuff if you have your .Minecraft not in the default location for example. You create profiles to access different versions of the game, management the JVM arguments which changes the way the game runs from using less/more RAM, to other technical things.
Other than the whole, Profile setup, for mods though after you have a setup with the required 'Modloader' (Forge, Rift, Fabric) you need a mods folder, inside that mods folder you put named folders like '1.5', or '1.13' (even though Forge understands mods file locations, Rift needed for 1.13 mods & Fabric needed for 1.14 snapshots mods don't notice these mods inside deeper folders, yet?)
I assume you mean 1.13 mods, but I understood modes so I don't know what you mean by that other than 1.13 on its own but the ability to have 1.13 and other versions of the game, meaning modes or profiles.
If you do mean 1.13 mods for a modded 1.13 environment though you'll need a thing called Rift, not Forge, not Fabric (for 1.14 mods), Rift.
Plenty is detailed here by myself since its similar to Forge but differs a bit in certain ways and is sort of a new-ish thing:
'Niche Community Content Finder' for fun and support of creators. I just swim through my 'Experimentation Pond' of content that intrigues me or really needs a bit of acknowledgement or bug fix testing/reporting. I also make modpacks/maps
Creator/Maintainer of Fabric, Rift & Tesla Mod Discussion Threads/Niche Mod Youtuber.