I think the construction is usually phrased: "given a line, to..", so it assumes a pre-existing line segment, rather then that being part of the construction itself.
I think it is possible that part of what’s missing in your “given a line to..” is that the description of the line in the .. refers to A and B, or possibility the phrasing is just informal.
Then to start the construction, you have to put one arm of your compass on the point A, and one on the point B. These points exist and are already marked, but you incur error when you place your compass down on each one. These are the represented by the blue and yellow circles in the diagram.
That line segment isn’t defined, it’s postulated (“there somehow is a line segment with endpoints A and B; you needn’t care how it came about”)