Attach headers to servo wires?
What I do in situations like that is get some of those breadboard hookup wires, like you have in your second photo, cut them in the middle (giving yourself two wires, each with a bare end, and a end with a plug on them).

Strip the bare end, and then solder it to the servo motor wires. Before commencing soldering, slip some heat-shrink tubing onto the wire. Once the wires are soldered together, move the heat-shrink tubing over the join and shrink it (with a hair-dryer, or heat gun).

You now have a plug for your breadboard which is designed to be mechanically strong, and the heat-shrink tubing over the join protects it from being flexed too much.

You could put some thicker heat-shrink over all three wires, to give additional stability.