The way I learned 'uter' and 'uterque' was as follows. 'Uter' is like the Greek 'πότερος', meaning (in interrogative uses) 'which, of two?' and (in non-interrogative uses) 'either, of two'. I learned to translate 'uterque' into English as 'both', but is this strictly accurate? When it is said that Cicero uses the method of argumentation 'in utramque partem', this clearly does mean that he argues on both sides, but is the more literal translation 'on either side'? If the former is correct, why is 'partem' singular and not plural?
Indeed, uter is a question word "which [of two]". And uterque can be translated as "both [of two]", but it might be better to think of it as "each [of two]". The reason is that uterque, like "each", is singular.
I think it is clearest to say that uterque is "each" and ambo is "both". The meanings overlap significantly, but there are two main differences:
- Grammatically, uterque and "each" are singular.
- Semantically, ambo or "both" refers to the two being together, whereas uterque or "each" means that the two are treated one by one.
When one is arguing both sides of a case, it may be best to do so in turn: first against one party, then the other. Being against both parties at the same time would be in no party's interests and would look more like arguing against the whole system or something like that. Therefore in utramque partem makes semantically more sense than in ambas partes.
When it comes to juridical oratory, arguing pro aliquo means defending and arguing in aliquem means accusing. Therefore in an oratory context I would translate in utramque partem as "against each party [in turn]". Translating with "both" is idiomatic English and fine, but I think "each" captures the idea somewhat more accurately.
Basically, you are correct, inasmuch as uter does indeed mean 'which[of two]?', and uterque can be translated as 'both'. The former is interrogative, the second is distributive.
Bear in mind that English has its own ways of expressing this kind of thing: for instance, 'did they both do it?' and 'did each of them do it?' and the replies 'yes, both did it' and 'yes, each of them/each one did it'. 'Both' refers to two acting in unison, while 'each' distributes the responsibility, not necessarily just to two. The difference from Latin is that in English 'each' is a general distributive, is mostly synonymous with 'either' and can refer to any number; while in Latin uterque is reserved for two.
I might also observe that we tend to be rather sloppy in colloquial English, relying on context to indicate the precise meaning — something that can become irritatingly apparent when translating into Latin.