It needs the cold timing advance to start. You can start the engine but without the timing advanced its hard on batteries. This thing apparently controlled it.
Not true, back in the day FoMoCo stated the cold advance and cold fast idle are strictly to lessen cold engine smoke complaints. Bought my pickup new in '91 and disconnected the timing advance about 2 weeks later. With the advance on, it sounded like the pistons were going to fly through the hood. Can't be good on an ice-cold engine. And with starting, glow plugs and fuel system in A-1 shape mine will start in 3 seconds at 10°-20°. According to the shop manual, the component you have circled is the low vacuum warning switch.
Heavy truck and diesel mechanic (thankfully retired after 30+ yrs)
'91 F-250 SC 4X4, 5spd. ATS Turbo, 3:55 diff.