Even after the update, I would like to know if I can still download the old version offline to play whenever I feel like it.

For the moment yes as they have not been updated, but the game will get replaced with the Re-Elected version.

Edit:

A Legacy Version of Saints Row IV will be also be available via a separate Beta Branch on GOG and Steam.

The older DRM free version is avaliable as a version update only via Galaxy.

So the offline file is still the old version? I ask this because one thing that bothers me is that many games that update, there is no way to download the previous versions. An example is Samurai Shodown V.

I want to have and keep for myself.

Currently the offline download is still the old one, yes. It was explicitly stated to be still available separately from the Re-Elected build after it arrives in this official FAQ: https://store.steampowered.com/news/app/206420/view/3646257001006919903

That says it only only be avaliable via Galaxy as an update.

It says no where that the GOG offline installers for the older version will be avaliable. Which seeing as the game has been replaced in my library I see no reason they will be when they get updated.

The offline installers of the classic versions will be removed.
You can see in gogdb's changelog that it already happened internally (not made public yet).

So create a backup copy now if you want to keep the classic version ("setup_saints_row_4_20170523_(12199)").

Note: The download site and link titles already say "re-elected" version, but when you click the links, these are still the old files so go go gol!

You are misreading it. It treats it as a different download just because the game entry is renamed.

Err ... no
Only the name is changed, the numeric and readable IDs are still the same. "removed" Really means "removed".
If it won't be added again, the installer will be replaced, the changes have just not gone public yet, that's all.
But go ahead, delete your offline backup and don't say I didn't warn you.
You will still be able to downgrade using Galaxy, so you won't lose the original edition for good.


On a sidenote: the readable ID of Jedi Knight is "star_wars_republic_commando_copy3", while Mysteries of the Sith has the ID "star_wars_jedi_knight_dark_forces_ii_copy3".
In these two cases someone messed up the first upload, the name was corrected later.

Renaming a product does not change it's ID or the file names. The name is just text info.

For the moment, the "Re-Elected" offline downloads still appear to be the "Game of the Century" version(s). I guess...grab 'em before Deep Silver nukes them too.

I'm saying that's why it comes up as a change on GOGDB. The actual files are still GotC Edition.

GOGDB has had hiccups before, where same downloads would be reported as removed and uploaded on the same exact day even though no changes actually happened on GOG's backend.

Game name changes don't cause that, you can check other games where this didn't happen, Jedi Knight for example.
And yes, the files available to the public are still original, I stated that a couple of times.

It's not the first time that a upload was not available to the public instantly.
In Destroy All Humans 2's case the change was on gogdb two days before release.
Star Trek Armada? The upload happened months before we got the game.
No one cared to rename the file at release, so it was still called "rc3". The installer was replaced or renamed some time later.


But in the end it doesn't matter. Who doesn't backup now, can always use Galaxy to restore the old version.