Yes. The BBC national on-screen "weather presenters" include both fully qualified meteorologists (with met degrees or similar who are also full Met Office forecasters) and less-academically qualified (though still Met Office trained) presenters. They all work for the Met Office rather than the BBC. I don't know if Jay is a full Met Office forecaster, or a presenter.
Historically all the Met Office weather presenters on the BBC were full forecasters, but this changed when News 24 launched I believe (back in 1997)
Just to pick up on your last point, when the Met Office won the BBC Weather contract, they noted that they (the Met Office) will supply all the data, on-screen forecasters and forecasting services.
The new contract means the Met Office will continue to supply weather services to the BBC. This includes the data on which the forecasts are based, on screen presenters and forecasting services across BBC output on TV, radio and online.