Where to stay in Mallorca for 6 nights?

I enjoyed stopping at Port de Soller on the north coast. Its an easy bus ride from Palma from the central underground bus station
You can also get there by vintage electric train to Soller and tram from there. Its a more expensive option but you should do the trip at least once
I hired a car from near the car park in Port de Soller for a couple of days
Cheap, not a new car, but cash deal and full insurance included

You're visiting Mallorca in June so it will be full of British and German tourists. Any of the places you've mentioned could suit the bill although Cala Millor, Cala Ratjada and Colonia De Sant Jordi might not have enough cafes, restaurants and bars that you are looking for although Palma will. There are enough authentic towns in Mallorca, in fact I can't think of one that isn't. The island is easy enough to get around so wherever you choose there are plenty of places within an easy drive that will occupy your time.

The two places to avoid are Magaluf and S'Arenal and you may also want to avoid Port d'Alcudia however the old town of Alcudia certainly has its charms.

I just spent six nights in Mallorca this summer. I would highly recommend renting a car for half of your time and spending the other 3 nights in Palma. With your car, make sure to drive along the western coast through the mountains, catching the towns of Valldemossa and Deia. You can stay in Soller, or get to it on the tourist train from Palma, which is probably easier since parking in Soller is very difficult. For more historic bases outside of Palma, try Pollenca and Alcudia (not Port d'Alcudia unless you want to be on the water). Arta and Capdepera are also very lovely and less touristy than the beach resorts, but smaller and have fewer restaurants and cafes. I would also highly recommend at least driving to Cala Sint Vicenc, the Mirador de es Colomer and Cala Ratjada. All are highly touristy, but the views are absolutely amazing!! I chose to stay at beach resorts outside of Palma because I love being near the water. They are all very touristy and will be packed with tourists in early June, but I knew that and made my decision anyway. I don't think you'll find any places on the beach that aren't just tourist zones. The authentic culture is inside the island, more away from the coast.

If you only want to have one other base outside of Palma, I think Pollenca or Alcudia would be best. The island is small, but there is more on that side to see/do than on the eastern coast.