The Cote d'Azur is definitely different than the rest of the "Region Sud Provence-Alpes-Côte d’Azur" (new name for the area encompassing five French departments). I'm not sure the Cote d'Azur (or at least the most eastern area that's now the department of the Alpes-Maritimes) was part of the "historic" Provence. But anyway... the main point is that it's very different and if you have only 10 days, maybe a different vacation.
There are cities and large towns in Provence: Marseille, Aix-en-Provence, Avignon, Arles. That would be one kind of vacation. You wouldn't really need a car... in fact, a car could be a complexity. You can travel between them by public transportation and also reach some smaller places by bus or train or take a day tour.
But I think to most of us who love Provence, we're thinking about the beautiful countryside, the mountains, the vineyards and olive trees, fields of poppies or lavender (depending on the time you visit), the stone villages, the limestone cliffs. Provence is a big area, and there are several possibilities of where you could base with plenty to do to occupy your time within 30-40 minutes. You can daytrip to one or two of the larger places by car or to another area. Our base is the northern area of the Luberon, but we also enjoy day trips to other areas. But other people enjoy staying on the southern side of the Luberon mountain, the area north around Mont Ventoux and the Dentelles mountains or the area around St. Remy de Provence and a little mountain range called Les Alpilles. And then there are the even more remote areas of Provence, but I think not for a first visit. All these areas have their fans, and I'm sure you could have a great time in every one. If you were more interested in wine or hiking or Roman ruins, that might suggest one area vs. another.
If you have 10 days, you could potentially stay one week in one area and maybe three days elsewhere. Many rentals are for a full week, but you might also find shorter rentals. I think there's a real value to staying one week in one place vs. moving around too much. We do love living right in the village where we can walk to the boulangerie in the morning (we have two) and to and from dinner at night... no worries about driving after dark after sharing a bottle of wine.
I remember my first time trying to decide on where to base in Provence. I had no idea about the area and I looked at so many different options I started to overthink it. We chose a house we liked in a village that had a boulangerie and then we planned our week around that.
Hopefully others will jump into the discussion--
Kathy