I think to avoid misunderstanding, instead of Crypto or Cryptocurrency, the term used to direct other cryptocurrencies should be altcoin or alternative cryptocurrency. Since Bitcoin is under the umbrella of the term cryptocurrency, it isn't wrong to say Crypto in place of Bitcoin, besides, Bitcoin is the icon of Cryptocurrency, IMO
The opinion you give is not wrong, but I personally always sort out between Bitcoin and other cryptocurrencies because for me Bitcoin is more than any cryptocurrency that exists at the moment and when someone says cryptocurrency without mentioning Bitcoin, I don't think that person is likes Bitcoin because it could be when someone says cryptocurrency is only for altcoins that he likes, not for Bitcoin which is clearly liked by many people.