to be alone: to stand apart from one's own kind, whether left solitary or broken away from the group
This verb turns up only three times, and each time it describes one living thing standing apart from the others it belongs with: a bird on a rooftop with no flock, a wild donkey off away from the herd, and a soldier who has dropped out of a marching column. The picture is always the same shape, so this is one idea rather than several.
What changes is whether the aloneness is suffered or chosen. In the psalm it is suffered: a sick man lying awake all night is like a bird with no companions, and the word carries the ache of being forgotten. In Hosea it is chosen: Israel has gone off to Assyria the way a wild donkey strikes out on its own, and here the aloneness is stubborn independence rather than sorrow. In Isaiah the picture is an army on the march, with nobody straying out of line.
The closely related noun (בָּדָד) covers the same ground on a much larger stage, since it carries the promise that Israel would dwell "alone" in safety and, in Lamentations, the reversal of that promise when Jerusalem "sits alone." This verb never reaches that grand language. It keeps its eye on one bird, one donkey, one straggler. A different verb again (בָּדַל) names the act of separating one thing from another, dividing light from darkness or holy from common, while this word names the plain state of already being on one's own.