Please help me understand how to wire a circuit for this welder
Correct.  This welder has given you a real &quot;jack-a-doodle&quot; set of specs that make no sense at all.  Running 120V/15A is only there so if you have light jobs, you can get 'em done using existing circuits... if you were wiring in a new circuit just for the welder, you would NEVER run 120V, since 240V is a possibility.
What's more, 30A+ 120V sockets and plugs are exceedingly rare and fairly useless for any other purpose.  They are only used for small travel trailers.
So here's what I would do if I were you.
Forget 120V. Go 240V if you are wiring a circuit.
I would use a 30A, 240V breaker.  (not least, 30A/120V breakers are rarely used and practically useless. Your local store's stock will be covered in dust.)
I would use 10/3 w/ground cable to bring that out to a NEMA 14-30 receptacle (standard 4-prong dryer plug) and wire that up all proper. (4 wires, neutral and ground separate).
Since the factory power cord is only #12, I would lop off the NEMT 6-50 plug and wire it to a NEMA 14-50 plug.  Don't use the neutral pin, you don't need it.
Why 120/240V
First and foremost, because presence of a 30A+ 240V socket in your garage, adds value to  your house.  Lots of home buyers these days are looking for electric car charge ports, so seeing that 14-30 ready to go will give you an edge with some buyers.
But also for $3 more in /3 wire instead of /2 wire, and probably $5 savings by buying a more popular socket, you now have plenty of power for the welder.  You're gonna have welding problems as you learn, but at least now, you won't have to worry about power being a problem.
An alternate plan
If you don't like having to change the welder plug, and still want that &quot;electric vehicle&quot; home equity bump, then do this, instead.

Fit a 40A breaker in that 40A subpanel. Really.  Trust me.
Run 8/2 wire.
Use a NEMA 6-50 receptacle. (really. NEMA refuses to define a 40A receptacle, so NEC just says use the 50A).

That was easy.
You still get to &quot;count coup&quot; on the electric vehicle capability, since EVSE's don't actually need neutral.
Yes, I know you have other loads in the 40A subpanel.  The 40A breaker is allowed because of the really weird provisioning rules NEC has for welders.  It gets into lots of complicated stuff about duty cycle.  The gist is your hobby welder doesn't actually need a full 40A provisioned to it, but it's allowed to use a 40A breaker and socket because when it's on, it can draw that much.
Also, let's talk about that #8 feeder wire to the garage. We might be able to bump that feeder up to 50A.