LED switch behaviour
The right side circuit is not usually a good way to wire an LED for the very reason you mentioned (unless that voltage drop is intended).  Also the diagram label is incorrect where it states "light on when open switch".  The LED will come on only when the switch is closed.

Based on the voltage drop you saw testing the right side circuit, it doesn't seem there is any series resistor, so you must include one to prevent burning out the LED.

I think this is what you are looking for, where the LED only comes on when the switch is on, but the load still gets full voltage.







simulate this circuit &ndash; Schematic created using CircuitLab