Transit visum or not?

Honestly speaking, you will never receive any clear answer from the airline. The only people who know about the rules regarding visa free travel within mainland China is airport staff. Airport staff at Phuket airport is well informed and I personally don't have any reason to doubt that they will allow you to travel the itinerary that is definitely eligible for 144 hours visa free transit in Shanghai. Of course, I hope that you have mentioned all your stops within your planned itinerary.

To conclude, it is totally irrelevant which airlines will you fly with and you really don't need any visa for your Shanghai transit. It is important to have with you your flight tickets in hard copy and preferably an accommodation booking in Shanghai. Please, keep in mind that you will not apply for any kind of visa upon your landing in Shanghai because you will fly visa free. You will just fill out a blue form at the relevant counter, they will put the sticker in your passport and you will leave the airport free to explore the area of Shanghai and it surroundings.