to seek: to go after something or someone you want, whether by looking for it, working to get it, or asking for it
At its core this word is about wanting something enough to go after it. The wanting is the constant; what changes from verse to verse is how the seeking shows itself. Sometimes it is literal looking, hunting for a lost donkey or a person who is hiding. Sometimes it is the steady pursuit of a goal you cannot simply pick up off the ground, like wisdom, peace, or favor. And sometimes the seeking is aimed at a person with hostile intent, as when an enemy seeks someone's life, meaning he is out to kill him.
The same verb covers turning toward God. To seek the LORD is not to search for a lost object but to orient your whole life toward him, to come looking for his presence and his help. The Bible uses this word for that pursuit again and again, and the warmth of it (longing, devotion, dependence) sits inside the plain idea of going after what you want.
A further shade is to request or require: to seek something from someone is to ask for it or demand it. Because the line between trying to get a thing and asking a person for it is thin, these senses shade into one another, and a few verses could be read more than one way. What unites them all is the forward motion of desire: the seeker is reaching for something not yet in hand.