The meaning of "to seek" in the Bible is pretty straightforward, but there is a bit of nuance. In most cases "seek" or "look for" is connected with truth and knowledge, where "desire" is connected with love and affection. This makes sense; the idea of "seeking" is connected to sight, and sight corresponds to our intellect and capacity to understand truth.

(Odkazy: Apocalypse Explained 550; Apocalypse Revealed 429; Arcana Coelestia 4719, 6770, 6771, 7021, 8298)

7021. 'For all the men seeking your soul are dead' means the removal of falsities endeavouring to destroy the life of truth and good. This is clear from the meaning of 'being dead' as the fact that they have been removed, since those who are dead have also been removed; from the meaning of the Egyptians, to whom 'the men' refers here, as those who are under the influence of falsities, dealt with in 6692; and from the meaning of 'those seeking the soul' as those who endeavour to destroy life. And since spiritual life is the life of truth which belongs to faith and of good which belongs to charity, it is called the life of truth and good. From all this it is evident that 'all the men seeking your soul are dead' means the removal of falsities endeavouring to destroy the life of truth and good. In the Word 'soul' (anima) is used to mean every living thing and is also attributed to living creatures (animalia). But the proper use of the word 'soul' is in reference to a human being; and when used in this way it can vary in meaning. A person himself is called a soul, because the life in general within him is called the soul; more specifically the life or activity of his understanding is called such, and so too is the life or activity of his will.

[2] But in the spiritual sense 'soul' is used to mean the life of truth belonging to faith and the life of good belonging to charity, or in general to mean a person himself in respect of his spirit that lives after death, which is the meaning that 'soul' has in Matthew,

Do not fear those who can kill the body but cannot kill the soul. Matthew 10:28.

In the same gospel,

What does it profit a person if he gains the whole world but suffers the loss of his soul? Or what will a person give as a price sufficient for the redemption of his soul? Matthew 16:26.

In Luke,

The Son of Man did not come to destroy people's souls but to save them. Luke 9:56.

In Ezekiel,

You have desecrated Me among My people, to kill souls that ought not to die, and to keep alive souls that ought not to live. Ezekiel 13:19.

In these places 'soul' stands for a person's spiritual life, the life which is that of his spirit after death. 'Killing the soul', 'suffering the loss of one's soul', and 'destroying the soul' stand for dying spiritually or being subject to damnation.