Esdra, you are onto something big, but not quite in the manner you are thinking.
Anglicans are not a “Church” since they lack valid orders, that is a Priesthood. They might have practices that, more or less, resemble Catholic practice and do (sometimes) claim to be a part of the ‘Catholic’ Church, but they most certainly are not.
Old Catholics are relatively few in number and came about when they rejected the Dogmas of the First Vatican Council. They did possess a valid Priesthood, but were in schism. However, it is unclear now if they currently have a completely valid Priesthood and many question whether their Priests perform valid consecrations of the Eucharist. They have become extremely liberal by doing things like embracing homosexuality and the concept of ‘women priests,’ an impossibility.
The Polish National Catholic Church is relatively few in number and came about over a jurisdictional squabble. They did have a valid Priesthood but were and remain in schism.
The Liberal Catholic Church was never part of the Catholic Church. An Anglican theosophist formed this sect and was ordained by a former Catholic priest who was consecrated a Bishop by the Old Catholic Church. (So already, you can see a very messy line of Apostolic Succession, but a valid one at least initually.) This sect holds extremely non-Christian beliefs such as reincarnation and theosophy.
So, with that in mind, there are different pieces of the one, holy, Catholic Church.
Eastern Catholics are Catholics who have their own rites and their own Church structure (this is known as sui juris.) They are distinct from Latin Rite Catholics, whose Patriarch happens to also be the Pope.
There are 23 sui juris Churches (the Latin Rite is the largest by far.) Within each sui juris Church, there is a diversity of liturgical rites (e.g., in the Latin Rite, the Roman Rite is the most common, but there is also for example the Ambrosian Rite.)
Eastern Orthodox Churches are valid Churches, but are in schism. Similarly, Oriental Orthodox Churches are valid Churches but also in schism.
I hope this helps!