That's right! I've already talked about this in at least one other thread (too much bother to find the link). The keyword, to find papers of this earlier "covariant LQG" is just the author's name Alexandrov.
Livine co-authored some with Alexandrov, but the later is by far the main and almost the only one who has worked on it.
Sergei Alexandrov started in 2001 working on something he called "covariant LQG" which goes thru much the same development as the old canonical LQG but uses the whole 4D rather than a 3D slice. A few other people got in briefly but it never caught on. Not much has been written about Alexandrovian quantum gravity since 2005 or 2006. Alexandrovian quantum gravity does not have an Immirzi parameter and does not get the discrete spectra for the area and volume operators.
So when Rovelli uses the term covariant LQG he means something quite different. It is a post-2007 thing that depends on some 2007 papers having proved that the new spinfoam models (which are covariant) reproduce central results of the old canonical LQG.
My impression of Alexandrov is that he is very smart, creative, and a bit of a loner. I have corresponded briefly with him back in 2003 when he finished his PhD at Paris and got the postdoc at Utrecht. I respect him very much. His Alexandrovian "covariant LQG" might in fact even be right (as an observer I cannot judge the odds) but it has not caught on with the rest of the community in all the time since 2001 when he presented it.
To me, Rovelli's usage of the term makes better sense. What Rovelli calls covariant LQG
1. has a lot of people working on it
2. recovers the main results of old LQG (which Alexandrov did not)
3. is covariant
4. has in effect converged and merged with old LQG as a research program.