AFAIK cpu60 is the Apollo 060 CPU command, and cpu060 in the Blizzard 060 CPU command. When those 060 cards came out the standard Commodore cpu command from OS3.1 could only understand the 68040 as the 68060 didn't exist at the time, so that's the story of the cpu commands! There are probably others on aminet but in my 3.1 system I still have the old 3.1 cpu command, but I have the cpu60 command to hand too in c: . This is perfectly normal and what both the Apollo and Blizzard software installers do (the install disk).
Because the output of your cpu command made no mention of extra 68060 features I therefore assume you are running an old (maybe 3.1) cpu command. Personally, I am not a 3.9 user so I have no idea what it shipped with regarding the cpu command. Also, it means I can't really help you much with 3.9 patches and such. I would guess though that 3.9 comes with its own 68060 library and 68060 compliant cpu command. Try typing 'version c:cpu full' and see what it says.
I am actually a 3.0 user as I reluctantly had to upgrade my roms to 3.1 for my FastATA to work (computer wouldn't boot otherwise, just hung on a black screen).
There's no need to update your system at all though really. If it isn't broke, don't fix it! I have a 1.3 A500 with internal 2.5" HDD...and do I want to update that? No way. I particularly like that one because it is old and I don't have to think about updates. I can make the best 1.3 install ever in the whole world (eventually).