Regarding vitarka and vicara, the Abhidharmakoshabhasya discussions in chapter 2, karika 33 are also relevant (since they come from the same northern tradition that the Agamas are from, and it is from about the same time that they were translated into Chinese).
Let me reproduce some of the content (from Gelong Lodro Sangpo’s translations of Poussin’s French translation of the Chinese) below:
The first part of Karika 33 says, roughly:
Vitarka and vicāra are the grossness and subtleness of citta.
In the Bhasya, a debate ensues on the nature of this Vaibhasika definition for vitarka and vicāra because some Sautrantikas say a mind cannot be both gross and subtle at the same time. The Vaibhasikas retort that it is possible, like a mix of water and vinegar. The Sautrantikas then put forth their own definition, that they are “constituents for speech” [vāksaṃskāras], and quote a sutra.
The debate goes as follows:
[Objection to the Vaibhasika presentation:] - How can initial inquiry (vitarka) and investigation (vicāra) be associated (samprayukta) with thought (citta) both at the same time? Can thought be gross and subtle at the same time?
[Answer:] - According to one opinion, (1) investigation is to be compared to cold water, (2) thought is to be compared to fermented curd which floats on the surface of this cold water, (3) initial inquiry is to be compared to the warmth of the light rays of the sun which shine upon the fermented curd. Because of the water and the sun, the fermented curd is not too melted nor too coagulated. In the same way, initial inquiry and investigation are associated (yukta) with a thought: thought is neither too subtle because of initial inquiry, nor too gross because of investigation.
[Reply:] - But, we would say, it follows from this explanation that initial inquiry and investigation are not grossness and subtleness of thought but the cause (nimitta) of its grossness, the cause of its subtleness: the cold water and the warm light rays of the sun are not the solidified state and the fluid state of the ghee but the causes of these two states.
Other objections are presented: - Grossness and subtleness of thought are relative [apeksika] factors. They involve many degrees: (1) [in regard to different stages,] the thought of the first meditation (dhyana) is subtle in relation to the thought of the realm of desire, but gross in relation to the thought of the second meditation; (2) in regard to the same stage, the qualities and the defilements can be more or less gross or subtle, since they divide into nine categories … Thus, if initial inquiry and investigation are grossness and subtleness of thought, we would be forced to admit that they both exist up to the highest stage of the realm of immateriality. But they have ceased from the second meditation onwards. - On top of that, grossness and subtleness cannot establish a specific difference: thus one cannot differentiate initial inquiry and investigation.
The definitions of (the Sautrantika):
According to another opinion, initial inquiry and investigation are “constituents for speech” [vāksaṃskāras]. The Sutra says in fact:It is after having initially inquired, after having investigated (vitarkya;
vicārya) that one speaks, not without having init1ally inquired, not without having investigated.The constituents for speech that are gross are called vitarkas; those that are subtle are called vicāras. [According to this explanation, by vitarka and vicāra one should not understand two distinct factors, but rather the collection of thought and thought-concomitants which provoke speech and which are sometimes gross, sometimes subtle.]
Can initial inquiry and investigation be simultaneous?
(The Vaibhasika:) - What contradiction [virodha] is there in the case where two factors (dharma)-the first (i.e., vitarka) being gross and the second (i.e., vicāra) being subtle-are associated with one and the same thought?
(The Sautrantika:) - There would not be any contradiction if these two factors had a specific difference or difference in type; for example, sensation (vedana) and ideation (samjna)–although the first is gross and the second subtle–can coexist. But, when considering the same type, two states, a weak state and a strong state, a gross state and a subtle state, cannot coexist.
(The Vaibhasika:) -There is a specific difference between initial inquiry and investigation.
(The Sautrantika:)- What is this difference?
(The Vaibha~ika.) - This difference is difficult to speak about [durvaca]; but it is manifested (vyakta) through the weakness and strength of thought.
(The Sautrantika:) - The weakness and strength of thought does not demonstrate the presence of two specifically different factors, since one and the same type is sometimes weak, sometimes strong.
According to another opinion-i.e., ours-initial inquiry and investigation are not associated with one and the same [ekatra] thought. They exist in turn.
The footnotes to this translation contain some interesting quotations from other works.
K.L. Dhammajoti comments:
according to the Sautrantika "vitarka and vicara are not real entities but merely two designations for the gross and subtle states of citta. Their difference is one of degree, not of kind. Vasubandhu explains that their operation is not simultaneous but alternate.
The Nyayanusara by Samghabhadra states:
[Objection:] Grossness and subtleness being mutually contradictory, vitarka and vicara should not be co-nascent in the same citta.
[Answer:] Although the substances of the two co-exist in one and the same citta, the predominance of their activities differs in time. Hence they do not contradict each other. It is like equal parts of water and vinegar in combination: Although there is equality in substance (i.e quantity), there is a predominance in activity [of the one over the other]. In a gross citta, the activity of vicara is impaired as a result of the predominance of the activity of vitarka; so that though [the former] exists, it is difficult to detect it. In subtle citta, the activity of vitarka is impaired as a result of the predominance of the activity of vicara; so that though [the former] exists, it is difficult to detect it.
One might argue that this analogy is not valid inasmuch as the activity of vinegar predominates at all times. This objection is not reasonable. I do not say definitely that the vinegar is to be compared to vitarka and water [to] vicara; only that which predominates in activity is like vinegar: Within the citta, whichever of the two dharmas, vitarka and vicara, predominates in activity, is to be compared to vinegar; that whose activity is feeble is to be compared to water. Hence it is not the case that one and the same citta is at once gross and subtle; as vitarka and vicara, though coexisting in the citta, operate at different times.
Louis de La Vallée-Poussin comments:
So, we can see here two divergent understandings of these terms in the northern tradition. Vaibhasika thinks that it refers to two qualities of thought or mentation which co-exist (i.e. grossness and subtleness).
Sautrantika sees them as specific mental activities which are associated with nama. In this Sautrantika understanding, before one speaks, first one applies the mind to a thought initially (this is a grosser activity, presumably because it takes more mental energy to direct the mind to some new object initially), then one examines said thought (a subtler activity apparently). Furthermore they do not exist in the same state of mind, they instead exist one after the other in turn (first you place, then you maintain that and examine).
Vibhasikas object to this because the first dhyāna is said to have both vitarka and vicāra (which they think supports their view), but Sautrantika retorts that this is only a general statement, in fact in any particular moment of the first dhyāna, there are only four factors. This is because vitarka and vicāra do not occur at the same time, and they are merely designations for two activities of the mind (and not two real independent existents).
IMO, this makes sense if you think of the meditative process as placing the mind on the object and keeping it there (and placing it back when the mind falls away etc).