Which astika schools mention those muktis?
These different types of muktis you have mentioned belong to the bhakti school of thought where God is considered to have a spiritual form with attributes (sakara and saguna). But because bhakti school of thought doesn't directly belong to the six philosophical schools (saddarshana), these concepts of mukti can be said to go under Vedanta astika school. It is because the Vedanta school has many more subdivisions like advaita, dvaita, etc. of which bhakti is a major part having been suggested by four different Vedanta schools. So the concept of mukti is central to the Vedanta astika school.
Are there more types of muktis?
The advaita system preached by Shankaracharaya is known as kevala advaita (pure non-duality). It's a system that preaches the path of knowledge. So as per this school there are these two kinds of muktis like jivanmukti (liberated while alive) and videhamukti (liberation without the body or after death). Scholars also mention many other types of muktis as per the doctrine they preach.
Which types of mukti's are common across various āstika schools?
Like I had already written in this answer to your question, getting liberation from this material mode of existence is the common objective of all irrespective of how they define what happens after emancipation. Astika schools like Samkhya and Nyaya give importance to the acquiring of right knowledge to cast out ignorance rather than to achieve mukti. Then in the Mimamsa school of thought attaining heaven is called as mukti. Hence, because mukti is the concept that belongs primarily to the Vedanta school, the Sayujya mukti where the jiva merges into the absolute losing its name and form is common to all types of Vedanta schools of thought. But it may not be the objective of all.
Is viśate also a type of mukti?
I am not sure where you read or heard it, but visate is a word which simply means "enters" or "goes into". So no, it's not a type of mukti.
Are sāyujya and kaivalya same, but different terms in different schools?
No Kaivalya and Sayujya are not same. The concept of Kaivalya belongs to the Yoga school of thought. Attaining Kaivalya is the end objective for them. It is a kind of liberation where the jivatma maintains its independent conscious existence and yet stays free from samsara. But in Sayujya mukti the jiva merges in the effulgence of the Brahman forever.
I hope this answers all the questions you asked in this post. You already have the Bhagavatam references from the yahoo question you mentioned. And in the Bhakti school of thought, Shrimad Bhagavatam is highly respected and also accepted as a source of valid knowledge.