If you cannot ensure that the rightful owner retains ownership, then the best choice is to do nothing. If you do anything, you are potentially harming someone, and if you do nothing you lose nothing, so the best outcome comes from doing nothing. If you take the coins (or "safeguard" them) you are stealing them since you don't know that they were lost or abandoned.
If you find the private key online like the question stated then if you do nothing -
With the key being online it will only be a matter of time when someone will either:
a) Steal it
b) Safeguard them
So doing nothing is just leaving the decision for someone else.
If you move them to another wallet for safe keeping then at least they don't get stolen and the owner has the chance to recover them.
Bitcoin is a messaging system (they can send a small amount of bitcoin with a message if they want to get in contact) and there would be various ways of proving ownership.
"Safeguard" is just another way to say "I might as well take them because someone else is going to steal them anyway." However you want to justify it, it is stealing. If someone is going to steal the coins, it would be better for you if it were not you.
Tell me, how are you going to contact the person that owns the address? There is no Bitcoin "messaging system". I don't know where you got that idea. The odds of you returning the coins that you are "safeguarding" is 0, so it is just stealing.