I remember from previous threads that the British Embassy doesn't require Thai-English translations of official documents to be authenticated by the MFA. But what actually counts as a certified translation? Does it have to have the stamp of a translation agency? Or is the signature, address, phone no. of the translator enough?

Most (all?) translation agencies will place a stamp on the translation which says in English and Thai that it is a true translation, signed by the translator.

This is perfectly acceptable not only to the British embassy but also in any situation once in the UK where the document plus translation is required.

The need to have an affirmation of freedom to marry stamped by the MFA is a Thai requirement, not a British one.

Most (all?) translation agencies will place a stamp on the translation which says in English and Thai that it is a true translation, signed by the translator.

This is perfectly acceptable not only to the British embassy but also in any situation once in the UK where the document plus translation is required.

The need to have an affirmation of freedom to marry stamped by the MFA is a Thai requirement, not a British one.

And how about an independent translator who doesn't have a company? Is a stamp required or is the signature enough?

Depends what you mean by an "independent translator." If it's done by you next door neighbour as a favour, don't be surprised if it isn't acceptable.

For a couple of hundred baht a page, it's worth having it done properly.