Yes, but that whole pesky, "Once alive" bit tends to mean that there's no life energy there.
And, I'll note that the lich doesn't actually have a soul in his body. He's got his in a jar somewhere. And, if you hit him with a True Rez, he doesn't become a living lich, he outright is destroyed.
Same as the vampire. The person reverts to however he or she was before she was killed by the previous vampire. Including alignment and whatnot.
Spirit and soul are not synonymous in D&D. Souls are specific in D&D canon. Lot's of Fey are nature spirits - not souls.