I just wanted to clarify, that in both Halliday's and Martin's interpretations, register refers to the relation between field, tenor and mode and their linguistic realisation. The difference is that Halliday defines the relation 'from below' as "the linguistic features which are typically associated with...particular values of the field, mode and tenor" (Halliday and Hasan 1976), while Martin defines it 'from above' as variations in "field, tenor and mode realised through language".
The difference seems to me not merely in terminology but in appliability, depending on one's goals. Likewise whether genre is construed as a more abstract level of context, or as a sub-class of tenor (functional tenor) or mode (rhetorical mode).
 In Halliday's model, the term 'register' refers to a point of variation on the cline of instantiation. Looked at from the system pole, each register is a subpotential of the linguistic system that realises a subpotential of the system of context; looked at from the instance pole, each register is a type of instance of the linguistic system, a text type, that realises a type of instance of the context system, a situation type.
In Martin's model, on the other hand, the term 'register' refers to a stratum. This means it is modelled as a system — not a subpotential of the system — that is more abstract than the content plane of language. However, 'context' does not have the same meaning in Martin's model as Halliday's model, since its contextual strata, register and genre, are both levels of language rather than a semiotic that is more abstract than language. As a result, the terms 'field' 'tenor' and 'mode' — at least to the extent that they are used self-consistently — also mean differently in Martin's model.
 As such, the difference between the models is not that Halliday "defines the relation 'from below'" and Martin defines it 'from above'".
 As such, the difference between the models is not a difference in terminology — "merely" or otherwise. The models do not use different terms for the same theoretical meaning (valeur); they (confusingly) use the same term for different meanings (valeurs).
 As such, not "likewise" for genre — not that an argument for genre as a stratum of context was actually made.