How do I register my EFIN to e-file 1040 modernized e-file returns?

Update your e-file application here: https://www.irs.gov/tax-professionals/e-file-provider-services

It sounds like you already have an EFIN issued by the IRS.  If that is the case, you will need to have that EFIN verified by Intuit before you can use it to efile returns.

You mentioned the instructions are not clear.  Did you refer to this article?

https://proconnect.intuit.com/support/en-us/help-article/account-management/register-efin-proconnect-tax/L9UQWErbz_US_en_US

It's a relatively straightforward process and my experience is that they would normally get it verified within hours, if not sooner, so long as you enter all the details correctly and the EFIN document you uploaded is current and acceptable.

If you were issued the EFIN within 12 months, you can upload either the EFIN letter or a printout of the Application Summary from the IRS e-Services website, which must show Approved, Active, and the relevant form series.  Otherwise, you must use the printout of the Application Summary.  See this article for a more detailed description and samples of these documents: https://proconnect.intuit.com/support/en-us/help-article/accounting-bookkeeping/submit-efin-registration/L1HoaIyXQ_US_en_US

When you update your EFIN address, you must make sure the address matches what is on your EFIN letter or Application Summary.  Otherwise, you will fail the verification.  Note that this is the address that will appear in the bottom of each tax return and is not necessarily the same as the address you enter separately under Firm Information.  Your firm's addresses under Firm Information and EFIN Verification are independent of each other - making changes to one will not update the other.

Hope this helps.  If you need more help, contact Support and have them walk you through the process.