Electronic – What’s the Common and the NO in this diagram

In the Nest Installation Guide, page 23, they show an installation example with low voltage. So I think they actually are mechanical relays, or at least relays that can actually work with DC.

So you can make it simpler: connect 1 or 3 to the raspi ground, and 2 to some raspi input with a pull-up, for example.

Google seems to be truly unhelpful on this one of their products.

Your existing thermostat has a live feed in at terminal 1, which is brown (as it should be) and the neutral is connected to terminal 2 (it should be blue, and is sleeved to indicate so), that is likely there for a function light. The switched output is at terminal 3, looks like that's a black wire.

So you need the brown to the C terminal and the black to NO.

Here's the only image I could find, from here

The E version is battery operated, so doesn't use any mains power. No ground connection or neutral are needed to the Heat Link, so those wires should be insulated and not connected.