1) They do not mean the same thing by ‘register’.
For Halliday, ‘register’ means a functional variety of language (with roots back to the Prague School and Firth).
Martin, on the other hand, equates ‘register’ with Halliday’s ‘context’, which removes the notion of register as a functional variety of language from the model. This is because a higher stratum is not a functional variety of a lower stratum; eg lexicogrammar is not a functional variety of phonology.
2) They do not mean the same thing by ‘context’.
For Halliday, ‘context’ is
- what people do with language (H&M 1999: ix),
- the ‘semiotic environment’ of language (p375),
- the “culture”, considered as a semiotic potential (p606).
Halliday’s ‘context’ is a level of abstraction that is realised by language.
Martin, on the other hand, in describing his model of stratification writes (1992: 496):
… the size of the circles also reflects the fact that the analysis tends to focus on larger units as one moves from phonology to ideology. Thus the tendency at the level of phonology to focus on syllables and phonemes, at the level of lexicogrammar to focus on the clause, at the level of discourse semantics to focus on an exchange or "paragraph", at the level of register to focus on a stage in a transaction, at the level of genre to focus on whole texts …
As this quote makes clear, Martin’s ‘context’ refers to levels within language, since 'a stage in a transaction' and 'whole texts' are units of language.