Normally with used phones, the warranty is only good for the ORIGINAL owner. Now the maker of the phone may allow their one year warranty to hold over for the the entire year, no matter who owns it. BUT I can see why they may only allow the warranty to be for the original owner, because neither they nor you, can know of any possible damage or modifcations done to the phone from the owner(s) that had it before you owned it, that may have voided the original warranty.
If they check and the phone has a version of software in it what would not have come in that phone you are SOL as far as warranty work and they do check.
Now I might be wrong, so....
Only way to know for certain, is to give Motorola a call, to see if warranty carries to the second or third owner of any given phone you are looking at. And of course you'll need some sort of proof of when the phone was first purchased, other than the sellers word that is.