I'm pretty sure 4.00 v 4.01 etc are the same thing. Ph varies with temperature so the ph difference is probably only due to quoting different temperatures (not that 0.01 is going to matter for brewing).
I would store in proper storage solution. I'm pretty sure it is a solution the same as what is inside the probe. Can't remember what it is though however potassium or something rings a bell.
Also, whilst I'm posting, a trap people fall into often is thinking that their new brute ATC ph meter corrects ph for measuring at mash temperatures. The ph of wort changes with temperature so you need to be comparing the ph at a set temperature. Room temperature (20-25ish degrees) is probably most common as measuring wort at mash temperatures isn't brilliant for the life of your probe.
Atc corrects for errors in the probe reading, not changes of ph in wort due to temperature.