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.
A recent trend in the HE encryption literature has been something called Somewhat Homomorphic Encryption (SHE). These are ciphers which support both addition and multiplication but not unlimited numbers of operations. E.g., BGN is a SHE cipher as it supports unlimited additions and one multiplication. The current FHE ciphers are all built using SHE ciphers plus a bootstrapping operation (or generating the keys such that enough of each operation can be supported to compute the desired function, making bootstrapping unnecessary).