The term Bhagavati certainly does refer to Thayar. Svami Bhasyakarar says ‘Bhagavati shrIyaM devI’ right at the beginning of Gadyam.
Bhagavati means one who possesses six auspicious qualities; namely – Gyana, Bala, Aishwarya, Veerya, Shakti, and Tejas. Even Emperuman is called Bhagavan because of the same, as one possessing six auspicious qualities; Bhagavan’s consort is Bhagavati, meaning one who possesses the same six traits.
The Bhagavati referred to in Kerala refers to Parvati Devi. They call her so out of affection. It is similar to how we out of affection name ourselves with Emperuman’s names. The term Ishvaran will only mean Emperuman, Lord of all universes. But people believe Ishvaran refers to Shivan. Likewise, though Bhagavati means Thayar only, Parvati Devi is commonly called Bhagavati.