You are allowed to file the current year being processed (2023) and two prior years (2021 -2022). Those are the three years that can be e-filed. Paper is not that much slower than e-file (according to what IRS publishes). 

You almost got the amendment rule right, but that is not the e-file rule. See page 3 https://www.irs.gov/pub/irs-pdf/i1040x.pdf