Does a French Pass Sanitaire based on a UK vaccine constitute an EU Digital Covid Certificate?

I'm a UK citizen and resident looking to travel to a handful of EU countries including France in the next few months.

The EU-wide Digital Covid Certificate should be valid as proof of vaccination in all EU countries, but people vaccinated in the UK aren't eligible.

France has announced they do accept UK vaccination certificates, and I was able to import mine into the French TousAntiCovid app and get a Pass Sanitaire. What's not clear to me: does this count as a full EU Digital Covid Certificate? If not, is it possible to generate one from TousAntiCovid, or obtain one any other way based on a UK vaccination?

This French government page indicates that the UK certificate is valid in France, but is unclear about other EU countries:

If you have been vaccinated in one of the following countries – European Union Member States, Andorra, Iceland, Liechtenstein, Monaco, Norway, Switzerland, United Kingdom (England and Wales only) – your country issues an EU Digital COVID Certificate which is accepted in France, or, in the case of the United Kingdom, a digital format also accepted in France. In that case, you do not need to convert your COVID certificate.