Open source has no formal definition. There are tons of different licenses. They can be OSI-approved or not.

Edit: I doubt the license for this code is an OSI-approved one with this registration business. But I have not checked and I won't check.

There is a trademark for "Open Source", and if you want to use it without written approval from OSI, you have to use it for something under an OSI-approved license: https://opensource.org/trademark-guidelines#Usage_That_Does_...

I think this is the closest you can come to a "formal definition", short of a law defining the term.

If you read the page you linked more carefully, you will see that OSI does not own a valid trademark for "Open Source", only for "Open Source Initiative".

OSI in fact tried to file for a trademark on 'Open Source' in 1999 [1], but failed because the term is 'too descriptive'.

[1] https://opensource.org/pressreleases/certified-open-source.p...

It's under EULA.