There's no distinction between a psychic animal and a non-psychic animal (such categorization is only for humans).
A psychic uses psychic powers through the higher functions of the brain. However, the ability to think can be a useful tool or an enormous obstacle: a psychic chooses to use it as a tool.
A psychic animal uses psychic powers through the lower functions of the brain. Psychic energies, senses, reflexes, subconscious decisions are all closely integrated to ensure the survival of the animal. We use the phrase 'animal instinct' rather than 'psychic animal'.
An animal psychic is a person who closely observes the psychic processes in animals and interacts with them for mutual benefit. The 'animal psychic' is a less powerful, modern day variant of the ancient 'shaman' who used and mastered the psychic energies of many animal archetypes.