On the 1DX, the custom presets aren't even selectable until you save something into the profiles, so no defaults at all. You even have to go into the menus to make C3 selectable...by default only C1 and C2 are selectable.
I noticed the inability to select other shooting modes while in the custom settings when I started using the 5D3. It surprised me and took a few minutes to figure out, so I am assuming it wasn't that way on the 5D2. I use the custom setting pretty much all the time during event shooting to swap between shooting with flash and shooting with ambient only...instead of making a bunch of changes to the flash, ISO, shutter, aperture, etc., I just make one turn of a dial. The 1DX has a button for it, which will take some getting used to, but it might even better, since I can now see the C number through the viewfinder.
IIRC, only C1 is on by default on the 1D X.
You've never been able to select a different shooting mode for a C# setting. On a camera with a mode dial, you're in P, Av, Tv, M, B, or a C#. Yes, they could make a firmware option, but they haven't so it's always been locked to the mode (P, Av, Tv or M) you picked when you first registered the setting. You can change just about anything else and re-register the same setting, but if you want to change C2 from Av to Tv, you have to start from scratch.
Since the 1D X is the first camera to offer C# settings that doesn't have a mode dial, it should have been pretty easy for Canon to allow you to change the base mode, but you still cannot.
Once you register settings to one of the C modes you can still make further changes once you are in that mode. You aren't locked to the settings you registered.
I use C1 for bird in flight settings and C2 for roosting birds. My C3 is still unused but I'm sure I will find a purpose for it soon.
Further changes except
the base mode.
FWIW, I like C3 for BIF - often, you want to get into that mode fast (shooting a perched bird which suddenly takes off, for example). With C3, you just spin the dial to the end without having to pay attention to the setting, no matter where the mode dial starts from (same logic on the 1D X, except there you're spinning the main dial, which doesn't stop, but still no need to pay attention, just give it a finger length's spin and you'll be on C3).