What you're describing is a milkshake, IMO.
Milkshakes involve ice cream. Not that I'm opposed to milkshakes made with coffee, but that's not what I'm drinking.
Nor does the SCAA agree with you. There's a whole range of terms for coffee drinks with various degrees of milk and/or water dilution. What I'm drinking is technically a flat white, since that's close to the ratio, and I heat the milk up in a microwave rather than deal with cleaning the steam wand.
No dilution: espresso.
1:1 espresso / milk: Machchiato. Technically supposed to foam rather than foamed milk but often isn't.
1:1 espresso / milk: Cortado. Foamed milk rather than foam. Pretty subtle distinction, if you ask me.
1:2 espresso / milk, but not foamed (or barely foamed): flat white.
1:4 espresso / milk: Cappuccino.
1:4 espresso / water: Americano.
1:5 espresso / milk: latte.
Really, what I'm drinking isn't remotely an outlier in the coffee world, or a "milkshake." The oddest thing I've done is make coffee using milk and ground coffee, no water, in a French Press.
That got me some "you're weird" reactions from coffee nerds.