Electrical – Making a momentary push-button switch activate a 5 VDC relay

Your idea 'current follows the path of least resistance' is an approximation, and it holds only when there is a limited amount of current. In your case you use a lab power supply, which is a constant voltage source, which will supply all the current you ask from it. Hence both the transistor and the red LED parts get current.

A way to switch the other LED of when you press the switch is show below.

^{simulate this circuit – Schematic created using CircuitLab}

The principle is that with the switch open, Q1 gets a base current via R1 and D1, so it conducts, and current flows through R2, D2, Q1, so D2 lights up. But the base current is too small for D1 to light up.

When you close the switch, Q1 gets no base current, and the current through D1 is large enough to light it up.

I don't know if this helps, but if you are looking to turn on the lights for a few seconds to a few minutes, you could use the button to trigger a 555 timer circuit that drives the relay. This one will turn on the lights for about a second. Increase R4 or C2 to increase the delay; there's a chart in the data sheet that you can use to figure out the right values to use. R2 will depend on your relay.

^{simulate this circuit – Schematic created using CircuitLab}