Not really his characterization, he has always been like that, unless the fundamental of "touch my son and you're dead" sounds to you like a strong front and not a softie who you shouldn't mess up with. His characterization has always been the same: a strong softie.
Also, this isn't just about Oden being dead. This is about Izou and Ace, their family and friends, and people Whitebeard himself knew and travelled with (I mean, Momonosuke was literally born in the Moby Dick, for example).
Him being like that is his characterization.
It comes with an asterisk, and not an absolute rule, yes. People might be disappointed about it, but this was already apparent when we saw the flashbacks pre-Marineford and saw WB more caring about Ace´s safety than taking revenge on BB at all costs, it´s consistent with that at least.
But they were all dead, that´s the difference, at least presumed to be.
WB´s sphere of responsibility, first and foremost, is his crew, nothing else.
And family is something you create in One Piece, not born into, i think this has been made abundantly clear.
And if we spin this even further, something that was stated several times, WB was also someone who always considered his role within the world and his function of basically holding the reins of the New World.
He mentioned this in Marineford, Doflamingo stated this in his throne speech, he was a fundamentally different type of pirate than Roger was.
So yes, i see a very clear distinction between "let´s go save our mate who is alive" to "let´s go avenge the dead".