"Spoofing" is making your system report a different System Version than you are actually on, usually the latest one. It doesn't actually change your System Version, so if a game specifically requires something, spoofing won't work. Its main use is to tell eShop that you're on the latest version so it doesn't block you.
Unlike the main game, there is no Universal/Legit version of the game update, so the only way to get one that works in sysNand without an exploit is to get it from the eShop. So, you use the guide you linked to spoof your System Version to access the eShop and download the update on sysNand. Once that's done, you don't need to spoof in order to run the game itself. The game doesn't care what System Version you're on as long as you have the features it uses. And as long as you have the latest update, you can play online without issue, even when not spoofing. You can cold-boot the system and run the game.
PastaCFW is a Custom FirmWare that includes patching signature checks, so it allows unsigned CIAs to run. I haven't looked too much into it, but I assume Ygw-Eshop Spoofer is something that makes it easier to spoof, probably like a standalone app. Since Pasta allows unsigned code to run, it's easier than relying on and using an exploit in the official system like the browser. (Though, getting Pasta running usually requires an exploit, such as Ninjhax, but once it's running, you're inside and don't have to worry so much.)
Also, if one of your main goals is to avoid having to use the browser exploit, you do have a couple options. On an Old 3DS, you can downgrade your system to 4.x using the Gateway launcher. On a New 3DS, you can install mset.cia (which is the System Settings app from the 4.x version). Both of those allow you to use the DS Profile exploit instead. By installing a specific set of data into your DS Profile (also known as NVRAM, since that's where the profile is stored), you can access the Launcher or Gateway Mode by going into System Settings > Profile > Nintendo DS Profile. Downgrading the Old 3DS does cause certain saved games to break, but that can be worked around by extracting saves first and then injecting them afterwards, I believe.