The youtube is showing a different (multi-panel) setup than the Sol-Ark diagram. The Sol-Ark diagram is similar to what I run - e.g. "just" the main panel with MTS. Here's my setup - the ATS = Sol-Ark for purposes of this comment. The Pro/Tran 2 is the MTS.
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>The above image is from the Sol-Ark manual. It shows a hot red wire going from a breaker in the transfer switch to the breaker in the main panel.
When you say "transfer switch" in this sentence - I'm not sure if you meant the Sol-Ark ATS or the Line/Gen MTS.
The MTS wiring to a main panel is completely different than running main panel power -> Sol-Ark (240/120v) and the Sol-Ark -> MTS (240/120v). The MTS wiring (in Sol-Ark diagram and in my system) is all about adding it's own breakers in a cooperative way with the main panel breakers... for several individual circuits... so you can choose Gen (power goes thru the MTS breaker) or Line (power goes thru the main panel breaker as it would have without the MTS). In my MTS case - 8 of the circuits are 120v only and 2 of them are tandem/240v. The wire coloring of an MTS not the same scheme as standard 240 black/red/white/ground
In the MTS case you bring in
1) A wire from the circuit of the main panel (by disconnecting it from the original circuit breaker in the panel)
2) Power from the ATS (either main panel or Sol-Ark/inverter)
3) And then hook them into the MTS - so the Line Switch position is powered by the original circuit and the Gen Switch position is powered by Sol-Ark ATS
4) Then run a wire back out to the original circuit breaker.
*Something like the above - there should be specific hookup diagrams for your situation / the MTS itself should have very explicit hookup diagrams.