So this leads to my question: Who gets to keep the original "BTC" token tag? exchanges should list the original chain as BTC, I hope they don't pull an "ETC" and pretend that the altcoin isn't the forked chain.
For me there could be potentially different approaches, but at the end I think the BTC tag will continue to be used for the "traditional" chain.
At the exchanges related to big-blocker mining pools (e.g. ViaBTC's exchange) there is a minimal probability that they will classify the "Bitcoin Cash" chain as "BTC" and the other chain perhaps as "Classic Bitcoin" or something similar (I think I remember they have called the "old" chain "Bitcoin frozen" in their announcement).
But as the Bitcoin Cash branding, from the start, is a bit different from the Bitcoin branding (they compare themselves to "Bitcoin", so they're saying they're not Bitcoin, at least not the "pure Bitcoin") I think most big exchanges will stick with BTC for the traditional chain. As big blocker exchanges have only a small market share, I think to not confuse their users, they will rapidly switch again to use "BTC" for "old Bitcoin" - they may not even try to change it.