(1) Correct - Sig 16 gives you two options - record one big stereo mix, or record one stereo instrument at a time.

(2) The artists you refer to might be sending multiple outputs from their mixer via cables to a separate multi-channel interface. Or they might record one track at a time.

You might as well send the stereo output from an analogue mixer to the interface you already own.

The options you listed before seem reasonable. Or consider going down the route of an analogue desk and separate interface and lots of cables!

Excellent, this is very clear.
Well, if I can record one big stereo mix with all my instruments or one stereo instrument at a time, it could be fine. I think it is gonna be my most common way of working. Playing more instruments and keep what is good only. Sometimes, I will need to keep everything, sometimes only one or two instrument.
If I have something interesting with multiple instruments with doing a jam session, I can also record each track separately and re edit then into the DAW the whole song.

Now I am considering two options:
- Soundcraft Signature 12MTK (support multi track recording)
- Soundcraft Signature 16 (doesn't support multi track recording) but it has more channels

The first is better for my budget. Is there a real advantage to have the multi track recording?