The Windows paging file (pagefile.sys):
Is used to store data that can’t be held by your computer’s RAM, which generally occurs when a program requests more memory than is currently available.
Is likely to be used much more frequently than the Windows hibernation file.
Can be moved to a different drive (as you've discovered).
The Windows hibernation file (hiberfil.sys):
Is used to store the contents of RAM when hibernating (hiberfil.sys is effectively a dump of all your current working memory).
Likely cannot be moved to a different drive (i.e. to the best of my knowledge, it typically needs to remain on the primary Windows drive).
Is it safe to do hibernation now?
With modern SSDs, it is unlikely that hibernation will adversely affect the drive (even if the PC goes into hibernation several times a day). As far as I am aware, with the total amount of data that can often be written to these drives, other kinds of mechanical failure are often more likely before a drive might even begin to develop errors.
How do I change the hibernation mode?
I am not sure what you mean by "hibernation mode". That said, you can turn hibernation on or off with:
Window 7
Control Panel\System and Security\Power Options
Select a power plan and choose Edit Plan Settings.
Click Change advanced power settings.
Look for the "Hybrid Sleep" option and toggle as desired.
Window 10
Control Panel\Hardware and Sound\Power Options
Click Change what the power buttons do
Click Change Settings that are currently unavailable (as necessary).
Check or uncheck Hibernate (Show in Power menu) as desired.
Note that you may also want/need to run powercfg -h off (Windows 7) or powercfg.exe /h off (Windows 10) from an administrative command prompt (Run → "cmd" → Ctrl + Shift + Enter).
Maybe there's a workaround?
I found this sketchy site claiming that you may be able to disable hibernation and edit the registry to change where hiberfil.sys is located but even those instructions state it may not work in all cases.