Garmin temp in runnig app

it does have an internal temperature sensor.  You will need to edit the Data Screens for the workout type.   See Garmin.com manual for a better writeup on how to edit the Data Screens for a given activityy.  or continue....

So...  RUN / options / Data Screens /  rotate to the screen you want and hit the EDIT pencil button or add a new Screen at the '+' icon screen.  

Once you select Edit a window on the right of screen will appear and scroll to Data Fields.  Move to the data field you'd like to change (will be blinking) hit the upper right button to select.  Then scroll through the categories of data options, to Temperature Fields.  Then the first one should be 'Temperature'  - click upper right button to select.

The problem is that when I select “temperature” (as you describe) the watch uses the internal thermometer, and not the external Garmin temp, even that is connected to the watch.

AAAH, gotcha.  I would try to see if there isn't a custom data field you can get from the Connect Store. Would show up in the ConnectIQ Fields section of watch when adding data fields.  I have been able to add an extra HR strap as a 'Auxiliary HRM' I ahve to imagine they would have similar for Temperature / Tempe

If the Tempe is connected, the watch will default to the Tempe reading, not the internal thermometer.  How do you know it's not showing the Tempe's reading?

there is also a widget in the CIQ store that will display BOTH internal and Tempe temperature readings.  You may find that helpful.

I think the external Tempe sensor only sends data once per minute - you may have to wait a bit for it to show up. 

Exactly as you say, the sensor works only during the activity and it does it very well, just enable the relative data field.