To clarify, with a clearer picture.
There are two separate temperature senders. One with two connectors is to the ECU.
One with a single connector is for the temp gauge.
The upper, usually brown or black sender has two active pins and is connected to engine management.
The lower, usually black or blue, sender has one active pin and is connected to the temp gauge.
The connector, single wire, you have mentioned is for the lower sender. It will show fullscale deflection on the gauge if earthed.