Is "Archi" simply a shorthand for a system of 5 staves for Violin 1 + Violin 2 + Viola + Cello + Contrabass? For example, can "Archi" be used when there is only one violin part (i.e. 4 staves)? Or is there another meaning besides that shorthand, such as returning to the "default" strings configuration (i.e. the same configuration in the first page)?
My reading is that archi denotes the "default" disposition of strings as established on page one. For a piece from the 18th or early 19th century, this would be four staves, as the cello and bass parts would share a staff. I suppose that this practice is dictated by the "house style" of the publisher that engraved the score. It's certainly not universal. To identify the parameters governing its use, you'd need to do a survey of different publishers' practices. It would be interesting to see whether this publisher ever did an edition of Brahms's Deutsches Requiem, the first movement of which has no violins, to see whether the practice of identifying the standard string choir is scoped separately for each movement or for the whole piece.
When did the practice start? What is it an abbreviation for, and in what language?
Divizna covered the language part as well as can be. As I imagine you've realized on your own, the idiomatic English translation would be "strings," even if archi literally means "bows." As to the time when the practice arose, I'd note that the practice of labeling staves on every page seems to have arisen in the middle or late 19th century; before that, they'd typically use the same disposition on every page of each movement, which makes for an inefficient use of copper and paper, but is somewhat less labor intensive because it requires less planning, and it is less ambiguous for the reader. Once you start optimizing systems by removing staves for instruments that have nothing to play, it gets very confusing if you don't label every staff. There are some related questions on this site, for example Partially-optimized orchestra scores?
I think it wouldn't be much for the music typesetter to label each individual staff with the instrument abbreviation in subsequent pages, so I wonder what's the rationale behind labelling the 5 stave-string section "Archi". Does it make it easier for conductors, so that by glancing that single word their mind automatically parse the 5 staves "in default mode" so to speak?
That's probably true (but remember that most scores from the 18th century or so to roughly the middle of the 20th are engraved, not typeset). Since the purpose of these abbreviations is to be read quickly, someone probably decided, and reasonably so, that a single word for the entire standard string choir is best, because it requires the least mental processing.
But why not for the other sections? The obvious answer is that the composition of the woodwind choir and brass choir are less standardized. From 17th century, arguably, but certainly from the 18th century to the present, the standard string disposition has been two violin sections, one viola section, and a bass part; then in the 19th century it became increasingly common to write the cellos and contrabasses on separate staves. There will hardly be any variation. For woodwinds and brass, variation is constant: two, three, or four parts for each type of instrument, different kinds of instruments within each family, and so on, even entire pieces being written without instruments from a given family (e.g., no trombones at all), and so on.
So this publisher decided to label the full string choir Archi in abbreviation. It would also have been reasonable to label each staff, and there are indeed publishers who do that. I even have a score of Beethoven's fifth on my phone that omits the label for the full string choir but does use abbreviated labels for the winds and brass: the purpose of the group of four staves at the bottom of the system, two with treble clef, one alto, and one bass, is so obvious that no label is necessary. Archi is indeed almost redundant, more a courtesy reminder than anything else.
What makes me wonder is why Rachmaninoff (or an editor) seems to label "Archi" only for the 2-1-1-1 segments of the score.
Because in those systems where the disposition is nonstandard, you need to tell the reader who's playing. The clefs and the bracketing do that in the examples you've selected for the question, but they won't always, and, besides, the bracketing is easy to overlook. The explicit label and the bracketing reinforce each other and help the reader understand as quickly as possible.
I was asking whether "Archi" conventionally can refer to any combination of the violin family once I establish the "default" scoring in the first page (say, 1 violin, 1 viola, 1 cello, 1 contrabass) rather than the more common 2-1-1-1.
I'm not sure you can say that there is any convention regarding the use of archi for this purpose. I've never noticed it before. If you did a survey of orchestral scores, you might find that not a single one using archi has a nonstandard string disposition, but then again it's entirely possible that the publishers who do or did use it never published (during the period when they were using archi) any orchestral music using a nonstandard disposition, so we can't "reverse engineer" their house style.
If archi makes you uncomfortable, just label all the staves. As mentioned, some publishers do this. If you like it, use it. If your orchestra has a nonstandard string disposition, you can use it for the full complement of strings, whatever that may be. See how your conductor or other readers of the score react. I'd bet that most won't even notice.
