Tier 2 cooling off period

If you were on a Tier 2 skilled worker visa but then quit your job and moved back to the US, what are the rules for coming back to the UK to work? Would I need to reapply for another work visa?

I've read there is a 12-month cooling off period, does that mean I couldn't reapply for a work visa for a year after I forfeited my last visa? And is it more difficult to be accepted for a work visa the 2nd time around?

If you were on a Tier 2 skilled worker visa but then quit your job and moved back to the US, what are the rules for coming back to the UK to work? Would I need to reapply for another work visa?

I've read there is a 12-month cooling off period, does that mean I couldn't reapply for a work visa for a year after I forfeited my last visa? And is it more difficult to be accepted for a work visa the 2nd time around?

The tier 2 visa has been replaced by the skilled worker visa.

https://www.gov.uk/skilled-worker-visa

The cooling off period no longer applies. And it does look like the criteria are easier than the old tier 2.

If you were on a Tier 2 skilled worker visa but then quit your job and moved back to the US, what are the rules for coming back to the UK to work? Would I need to reapply for another work visa?

Yes. You would need an employer to sponsor you for a visa. Your visa was linked to the employer who sponsored you. When you quit your job, your sponsor was required to contact UKVI within something like 10 days, to state that they no longer sponsored you, so that your visa could then be cancelled.

I've read there is a 12-month cooling off period, does that mean I couldn't reapply for a work visa for a year after I forfeited my last visa?

As larrabee has said, this is not a requirement of the new Skilled Worker visa. The UK brought in new immigration rules when they ended the EU's Free Movement of people to the UK, at 23:00 31 December 2020.

Before that-
The Tier 2 (Intra-Company Transfer) visa did not lead to settlement and these had to serve a cooling off period of 12 months outside the UK before they could apply for a Tier 2 General visa (which did lead to settlement). 
The 12 month cooling off was only for  the Tier 2 (General) visa  holder if they applied for another Tier 2 (General) visa from outside the UK. If they applied inside the UK, there was no 12 month cooling off period and these did not stop their 5 years to ILR clock. e.g. lost their job (and sponsor) and could not find another sponsor before their visa was curtailed or expired, they had to leave the UK and serve the 12 cooling off period before another Tier 2 (General) visa.

And is it more difficult to be accepted for a work visa the 2nd time around?

Makes no difference if you have the skills employers wants. The new immigration system is much better for skilled non-EEA citizens.