Your explanation of difference in meaning have sense. I really didn’t pay attention on this at all till now and always used variant with capital "B" that is "Bitcoin" in meaning "the currency". If to look through article about BTC on Coinmarketcap (https://coinmarketcap.com/currencies/bitcoin/) you will find out  that they also use variant with a capital "B" in meaning "the currency" as a synonym to BTC, or better to say as a proper name of this cryptocurrency as well as all other crypto: Ethereum, Tether, Dogecoins, etc.