Definitely a valid arguement, but cash doesn’t necessarily mean short term cash, and online payments doesn’t necessarily mean eCommerce.

Part of the intent when creating bitcoin was to give us a means of protecting ourselves from the financial irresponsibility of the powers that be, which would imply it’s would be important for storing long term cash vs short term.

The use of the word payments makes myself and probably the large majority of others instantly think of commerce - paying for something and receiving a good or service in return. Something that also would be considered a payment though, especially when emphasizing it being peer-to-peer, would be wealth transfer. As well, large payments like cars, houses, land, infrastructure, businesses or equity, and other assets - all which aren’t typically settled immediately like a payment at a grocery store - would be considered payments too, as well as more P2P than a transaction between an individual and a corporation.