What are the main types of fluency that are relevant to language learning?

Question

Fluency is often confused with proficiency; it has even been an issue on this site. Simply put, proficiency is the level you have reached, whereas fluency is based on the automaticity and smoothness with which you can perform certain language tasks. (See also Fluency on the University of Oregon Center on Teaching and Learning website.)

For example, people with Broca's aphasia speak haltingly (i.e. not fluently), although they still retain their overall language proficiency.

Also, in 1974, E. Hatch ("Second language learning - universals" in: Working Papers on Bilingualism, 3) made a distinction between "data-gatherers" and "rule-formers". Data-gatherers tend to focus more on the development of fluency rather than accuracy, while rule-formers adopt a more analytic, rule-based approach. This can lead to differences in fluency between language learners who have the same overall proficiency level.

In order to avoid future confusion on this site, I would like the answers to this question to list the main types of fluency that are relevant to language learning, ideally based on authoritative sources (e.g. scientific literature).

Update: For the sake of this question, I will assume the following definition of fluency from Analysing Learner Language by Rod Ellis and Gary Barkhuizen (Oxford UP, 2005), which Scott Thornbury quotes on his blog:

the production of language in real time without undue pausing or hesitation

Answer 1

Scott Thornbury has an interesting entry about different ways the term fluency has been understood:

[...] fluency is one of those elusive, fuzzy, even contested, terms that means different things to different people. In lay terms, a “fluent speaker of French” is probably someone whose French is judged as accurate, easy on the ear, and idiomatic. The term, however, was co-opted by methodologists (especially those aligned to the communicative approach) to describe the purpose of classroom activities whose focus is on communicating meaning, rather than on the practice of specific (typically grammatical) forms. Thus, Brumfit (1984) said that “the distinction between accuracy and fluency is essentially a methodological distinction, rather than one in psychology or linguistics”. And added, “fluency…is to be regarded as natural language use, whether or not it results in native-speaker-like language comprehension or production”.

[...] Ellis and Barkhuizen (2005), following Skehan (1998), define fluency as “the production of language in real time without undue pausing or hesitation.” That is to say, it is a ‘temporal phenomenon’. [...] Confusingly, though, they go on to say that “fluency occurs when learners prioritize meaning over form in order to get a task done”. This to me seems patently false: getting a task done is no guarantee that there is no “undue pausing or hesitation”. On the contrary, the effort involved in performing a task may actually increase the degree of dysfluency. And nor is attention to meaning a pre-condition for pause-free production. Some of our most fluent productions, as proficient speakers, are texts that we have committed to memory (tongue twisters, nursery rhymes, prayers, oaths of allegiance, etc) that we can trot out without ANY attention to meaning.

[...] In fact, it may be that fluency is indeed a function of memory, and that the capacity to produce pause-free speech in real time is contingent on having a memorised bank of formulaic language “chunks” – a view that was first put forward as long ago as 1983 by Pawley and Syder in their seminal paper, ‘Two puzzles for linguistic theory: nativelike selection and nativelike fluency’.

[...] Harold Palmer, as even longer ago as 1925, identified ‘the fundamental guiding principle for the student of conversation’ as being “Memorize perfectly the largest number of common and useful word-groups!”