From your link, it looks like "Name" is in Track 1:

  Start sentinel — one character (generally '%')
  Format code="B" — one character (alpha only)
  Primary account number (PAN) — up to 19 characters. 
  Field Separator — one character (generally '^')
  Name  — 2 to 26 characters
  ...

Thanks, not sure how I missed that.

It's also part of the EMV Specification and can be read with any nfc or smart-card capable reader. You don't need a special payment terminal.

https://www.emvlab.org/emvtags/show/t5F20/

There is one store in my neigbourhod where the payment terminals show this field (my full name) on the screen during checkout. I was very surprised when I noticed this the first time.

But apparrently not all card issuers fill the field with correct data. One card, a prepaid Visa from a big fintech, has "N/A" programmed in the field.