Homomorphic Encryption (HE) which supports any function on ciphertexts is known as Fully Homomorphic Encryption (FHE), while Partially Homomorphic Encryption (PHE) includes encryption schemes that have homomorphic properties, with respect to one operation (e.g., only addition or only multiplication, but not both).
It will depend on the exact system.
The cryptosystem of Castelluccia is additively homomorphic and is thus a PHE cipher.
The cipher of Xiao, et al. is fully homomorphic.
To determine if a symmetric scheme is a PHE or FHE, just look at what operations it supports. If it supports a single operation (say addition), then it is PHE. If it supports both operations (addition and multiplication), then it is FHE.