Airbus has significant (at least 25% last I looked) state ownership by various European governments. They did the development, the launching is done by Arianesp

Airbus has significant (at least 25% last I looked) state ownership by various European governments. They did the development, the launching is done by Arianespace which is half owned by Safran (which is also part owned by France). That combined with the Ariane programs themselves being initiated and funded by the ESA makes it a much weaker claim to be a commercial rocket launch. It's going to be hard to find a space rocket company which has 0% state involvement (e.g. Isar Aerospace and SpaceX have also accepted some amount of funding through venture rounds or contracts one way or the other at some point which can be traced back to governments, or purchased use of government launch facilities) by nature of "why wouldn't you accept a government launch at some point if you've already built a cost efficient rocket" or the like... but I think it's fair for Isar to claim to be launching the first commercial European space rocket since the program to build the rocket wasn't initiated by a state space agency or fulfilling a state agency contract by doing the initial flight.