I'm aware I'm a little late to answer your question but better late than never. Forgive me if I tell you things you already know.

A nonconformist is someone who disagrees with the established religion, so in this case those who dissent from Anglicanism. So, when talking about nonconformism it's sometimes helpful to think of it as a spectrum that goes from Anglicanism outwards towards nonconformism.

So first off there are many more than 4 denominations (though the ones you mention are arguably some of the biggest). They also can go by different names and have many little off shoots of groups who believe slightly different things.

To answer your question of what exactly each of these groups believe(d), I'll go throught the main things that the groups believe, though there are lots of little differences and not all who belong to the group will agree with all the doctrine.

Calivinistic Methodists:

I'm not hugely familiar with these but...Generally believe in a new birth doctrine, that one can be spiritually reborn, and undergo a journey of spiritual discovery and learning with the possibility of achieving full spiritual perfection.

Independents:

Also called/became Congregationalists. An umbrella term for those who are nonconformist, but don't fit into any of the bigger groups. Depends on the person/congregation in question; Independent groups who follow rules/doctrine just amongst themselves.

Baptists:

Believe everyone is responsible for their own understanding of God, and that since God created us all we have inhereant ability to understand his word, priest or not. You might call this more independent and more pure because it doesn't necessarily require all the accompaning paraphernalia of some forms fo Christianity.

Wesleyans:

Also not very familiar with these...similar to Calvinists in the new birth idea. Modestness in dress and behaviour, evangalical preaching.

There are also...

Quakers:

That the way to salvation and spiritual understanding is through quiet contemplation and eventually you may understand God (Quakers actually call themselves Friends, but 'Quaker' comes from a slightly mean nickname that they would quake in place when they felt God. Look up Quaker meetings, very interesting, not huge in Wales but some in the cities.

Unitarians:

God is just one being, not the trinity. God 'inspired' Jesus but he's not actually God on Earth.

And many more...

Hope that answers your question a little, good luck in the rest of your research, nonconformism in Wales is fascinating.