You are 100 % correct that this is how Chazal and Rishonim and Acharonim ultimately represented it.
However, see the sefer מה נשתנה, from רב דוד הנשקה, pp 125-165.
He meticulously shows there how Chazal’s representation of daled kosos (and just about everything else regarding Leil Pesach) evolved over time.
My representation was of how, according to him, it was ORIGINALLY understood.
Your representation is of how it was ULTIMATELY understood.
His sefer is a dense and scholarly work and simply summarizing won’t do it justice.
One needs to see how he builds his arguments though meticulous examination of hundreds of sources.
If you’re interested in the challenge, ayein sham.