So, I just wondered if Bach did put figured bass in any of his scores?
Or in the continuo parts?
Yes, frequently. There are many examples of continuo parts with figures. They were not generally copied by Bach himself, but they were frequently prepared under his supervision and used for performances that he directed. Presumably the parts that lack figures are more likely to have been used in performances where Bach played continuo.
You can see this on IMSLP by clicking the "parts" tab under "sheet music"; many of Bach's pieces have sets of parts that were prepared for performances during his lifetime. Sometimes Bach added figures in his own hand to a part that had been copied by someone else (see https://global.oup.com/us/companion.websites/9780190936303/res/app2/):
A manuscript entirely in his hand is a holograph, but many copies made by students or assistants include autograph titles, corrections, or performance markings (such as continuo figures) in Bach's hand.
You further ask:
Or would the continuo player have used the full score, written the figured bass in themselves, or just listened to what was going on in the other parts?!
No. Probably not, as there are too many page turns. But it's worth noting that the absence of figures in the score can easily be explained by the fact that if a keyboard player were playing from the score, perhaps as the conductor and possibly not the only continuo keyboard player, it would be possible to identify the correct harmony by looking at a few of the other staves.
The main reason I'm interested, is because I presume the figured bass in the score I link to above has been added by the editor ...
The link is now dead, so I am not sure what edition you're asking about, but the figures almost certainly come from study of historical sources rather than the editor's imagination. The parts available on IMSLP are very thoroughly figured, and although they date from several years after Bach's death, they were probably prepared from parts created during Bach's lifetime. One would have to be slightly skeptical of them, therefore, but far less so than of figures composed in the 19th or 20th century.
The edition should have an accompanying text that explains where the figures came from, however, so there should be no need to guess.
... and due to the chromatic alterations in the instrumental parts could have been interpreted in a number of slightly different ways.
I don't understand this at all. Do you have an example? Generally the correct notes for the figured bass are definitively determined by the notes being played by the orchestra or sung by the choir.
Would baroque compositions have had a "definitive" figured continuo part?
Some do, some don't (to the extent that any music is definitive, given that the composer might change it slightly, or sources may differ for other reasons). It seems that the Matthew Passion does.