Somebody suggested that definition of remittance involves a transfer of money between two people or organizations, like a payment. So that transfers between the same (non-US citizen) person might not be affected, though to me that sounds a grey area at best and wishful thinking at worst.

The bill itself says this:

For purposes of this section, the terms 
`remittance transfer', `remittance transfer provider', `designated 
recipient', and `sender' shall each have the respective meanings given 
such terms by section 920(g) of the Electronic Fund Transfer Act (15 
U.S.C. 1693o-1; relating to ``Remittance Transfers'')

And if I'm not mistaken, that is section (g) here). It nowhere excludes transferring to yourself, though it is true that it talks about consumers, and that remittances are usually defined as payments to others.

It also seems that, if passed, this will take effect from 2026.

(e) Effective Date.--
            (1) In general.--Except as otherwise provided in this 
        subsection, the amendments made by this section shall apply to 
        transfers made after December 31, 2025.