to transplant: to set out an already-rooted shoot or young tree in a chosen, well-watered place
This is a poet's word for planting, never used of ordinary farming. It pictures the careful work of lifting a young, already-growing shoot and setting it into a prepared spot, almost always one with water nearby. That is what sets it apart from the common verb for planting: this word leans toward deliberate placement, the kind of move a gardener makes on purpose so the tree will take deep root and last.
Every one of its ten appearances is a picture rather than a description of real gardening. The thing planted is always someone: the faithful person rooted beside streams of water, the worshiper flourishing in God's house, Israel as a vine moved into rich soil and later dragged off into a dry wasteland, and finally a tender shoot that God himself promises to set on his holy mountain. In most of these the planter is unnamed but understood to be God, and in the two places where the planter is named outright it is God speaking in the first person.
Because the same picture runs through all ten verses, this is best read as a single idea worked out in different settings, not as several separate meanings. The one motif underneath them all is that God's people are not self-rooted; they are placed. Their fruitfulness depends on where they have been set and on whether they stay faithful there. The very word that promises a restored future for David's line is also the word that pictures that line uprooted and exiled, which is exactly the double edge a rare and vivid term can carry.