Is my understanding correct that even with full support, the network might not change anytime soon, or at least until large operators start converting to segwit2x wallets?
Both biggest cryptocurrency exchanges, Coinbase and Bitfinex, have already confirmed that they are going to launch the SegWit protocol on their websites. This is confirmation that the market is ready for the new technology:
Bitcoin Core 0.16.0 Introduces Full Support for SegWit - Crucial Milestone in Bitcoin’s HistorySegWit and SegWit2x are not the same thing. the names are clearly different so are the projects. SegWit2x was an old proposal which shared the first part (being SegWit) with SegWit and then it had a second part which was the 2x part (being the increase of block size to 2 MB). it gained support for the first part and activated it but didn't gain enough support from community for the second part so it never was activated and the proposal was abandoned for now.
the news you are quoting is simply saying the
full support for SegWit is now added in one of the bitcoin clients called bitcoin core. this has nothing to do with SegWit support itself, it was active ever since September 2017 and it certainly has nothing to do with SegWit2x.