The following scenario is a bit simplified, but it does explain what I need to do.
What I want to do is select points 1 and 2 on the purple path in order to move them.
If I use the Direct Selection Tool, draw a rectangle around the points and then drag them, then points 3 and 4 get moved too. I don't want to touch points 3 and 4.
I know a possible solution is to double-click the purple line and then select the points. But in reality it's not just one purple line, it's many lines, and they are not grouped. I'm drawing a bus map, so different lines cross in different places, and I don't want to start grouping bus lines together.