To add a vertex you need to select the feature you want the vertex added to. Then activate vertex tool:
When hovering with your mouse over your selected line you will see a small plus sign between two verticies. Click on that to add a new vertex, and drag & drop that new vertex to the existing one of the other (not selected) line. With snapping enabled (by pressing s) the to verticies will be perfectly aligned!
Edit: As requested in the comments, to add verticies to all lines placed ontop of each other you have to enable topological editing:
Then select all the lines you want to add the new vertex to and follow above description (hover over the line segment, click the little plus sign which should appear and drag & drop the vertex to the existing one)