Should I use caulking or silicone to seal around a new door frame of exterior stucco house?
So... the word "caulking" refers more to its use/application as a linear gap sealer than it does to the material composition; Caulk can be made of most any material (including silicone), it's still called caulk.

Yes, as was your intuition, you need paintable caulk. Some silicone-based caulks are advertised as "paintable", but for myself I am dubious of the claim because I fear that over time the bond of paint-to-caulk will fail. For paintable applications I use a high-quality paintable latex caulk.