I was thinking of the more classical scenario when there's the original chain going on without any modification, and a new chain, which forks off with a new protocol. Then it is pretty clear that the original chain is, well, the original one.
Yes, here I agree - and in this case I think you would be right with your assumption that at least a very large part of the services would stay available to the users of the "original" chain.
Of course, if, at the moment of split, the original chain simply stops, and two NEW chains emerge, both with a different protocol from the original one, yes, then of course you get a confusion. But that can only happen if the system is *entirely* centralized, where almost all entities in the system have colluded into one camp or another one.
I think I have overcomplicated things with the UASF/"Barrycoin" scenario. But my scenario with "splitting services" would also be adaptable to a classic scenario with one hard-forking party where the hard-forking chain has the backing of large parts of "the economy". Think of the UASF. I could imagine a scenario where their supporters could get more than 50% of the economy and a substantial portion of the miners to support a chain that eventually becomes incompatible with the "old chain".
The old chain would become then the minority chain. In a case of a 60%/40% split it would continue to be usable, though - but several services would cease to exist for the old chain because they support the changes the upgrade introduces (e.g. Segwit). Some could offer their services for both chains, but I think that would be a minority.
I was thinking of the more "decentralized" approach when there are no "agreements in meeting rooms" at any point, but that suddenly, an entity proposes a fork as individual initiative, the rest of the system not even knowing about this plan until it is announced.
I think a smart "forking group" would announce their plans well in advance to try to get adoption by the economy and miners. BU made this exactly and failed, but another group may have better luck (see Ethereum).
It wouldn't be any different than coinmarketcap. A fork is just a new crypto currency, with the funny property that if you were holding coins on the original chain, you also happen to hold coins on the new one, coins which you may care about, or not. You just get a holding on a new chain if you care to download the wallet. You might just as well not.
This is no different than if one would tell you that you can download, say, a litecoin wallet,
That would only be true in your case where a fork is a minority option not announced well in advance and without substantial community support. Cryptocurrencies are social communities where some kind of "valuable token" is accepted. If the community splits in two parts that are roughly equally important (50/50 to 70/30, for example) then you do have the confusion which chain is the "original one" or the "most supported one". And then you would have all the usability hassles I'm talking about.