Now from register (axiologised field)…
The task for functional grammarians is not simply to classify types of grammatical features, as you say, it is 'to figure out the implications both for theory and for descriptions’. I’m not sure a 'form/function’ contrast is adequate for this task. Rather we need to tease out functions at the strata of grammar, discourse semantics, and register, treating each as a layer of meaning, as Firth foresaw.
 In SFL, and in the field of linguistics generally, registers are functional varieties of language (not context). In SFL, field is the ideational dimension of context (not language) — context is realised by language.
Registers are varying subpotentials of language that realise situation types (varying subpotentials of cultural context).
Field refers to 'what is going on' — in the cultural context — when people are 'languaging': speaking, listening, writing and reading. For example, it is 'what is going on', in terms of the culture, when you are reading this.
Axiology is the philosophical study of (mainly) two kinds of values: ethics and aesthetics. The term was first used by Paul Lapie, in 1902, and Eduard von Hartmann, in 1908. [Cf. judgement and appreciation in the interpersonal system of attitude in Appraisal Theory.]
 Firth didn't foresee register as a contextual stratum in SFL. Firth was a first-rate thinker who valued intellectual rigour very highly.