There are a couple of points to make here LV4. I think you ask some extremely good questions!
First of all, there's no avoiding the fact that there is domain specific information here. Which is to say that in this day and age, although some folks do still work on their own cars (me for example on an older car of mine), the majority of folks just don't do this that much any more. I have the idea that a lot of English speakers and listeners incorporate this domain specific information when speaking or hearing this sentence. The simple fact is that there is a very strong probability that my 60 year old father wasn't under the car replacing the transmission - far more likely that he took it to a mechanic who actually performed the work.
The next point is that there is a slight clue in the use of the word "repaired". Again, I'm speaking of probabilities, not certainties. But there is an oh so slight sense when the word repaired is used that supplies the connotation of a more formal process. Don't forget we have another word i.e. "fixed". As you probably know, an anglophone is often aware of the two words for each situation and the choice of word in some cases does convey information. If my father had worked on the car himself, I would much more likely have chosen the word "fixed" or "worked on" or "spent the day working on" (things don't always go so smooth when your doing the work yourself
). This distinction
is not one that anyone could absolutely rely on, but it is a shade of meaning that exists nevertheless. I might choose to "respond" to a letter from an attorney but if you were my personal friend I might "write you back". So as English speakers we are often subconsciously aware of the 2 words that mean the same thing and choose the one that fits the situation best - kind of like "tu" and "vous" but applied to more words
Oftentimes the more formal word is Latin or French, while the less formal word is Anglo Saxon (like the "respond" and "write back" example). In this case, "repair" and "fix" both exist in French, although "fix" in French means only one thing (stationary) while in English it has two meanings (stationary and to repair). So while "fix" exists in French, it's meaning "to repair" does not. And this word "sounds" more Anglo Saxon to me than repair. So the rule that the Latin word is the more formal doesn't strictly apply here, but it partially does.
I think there may be one more nuance in addition. In most of the cases, the cues I mentioned above are sufficient. But also in spoken English there is the opportunity to stress certain words more than others - something that doesn't translate well to a written format such as this board. If, in spoken English one wished to erase any ambiguity I think one would say something like
This is the car that my father
had repaired yesterday
i.e. stressing
both words because essentially both verbs are important - the verb of having something done and the verb of repairing.
vs.
This is the car that my father had
repaired yesterday.
In this case we only stress the "repaired" verb because the other verb is just an auxilliary.
Also if context and stressing syllables aren't enough we can also say things like "had had repaired" or "had gotten repaired" to further eliminate ambiguity.
So I think we have domain specific knowledge (a lot of people don't work on their own cars), choice of verb (repair vs. fix), which words to accent in speech, and finally using extra words like ("had had" or "had gotten"). At some point the listener is going to arrive at the correct meaning, one would hope!
