You're right: the "more distant galaxies are moving away faster" point is just Hubble's original observation of an expanding universe. It's not an argument for cosmic acceleration. (If you see people making that claim, they're probably either speaking carelessly or not experts themselves.)
The conclusion that the expansion is accelerating was a quite recent result: 1990s, I believe. It's based on careful measurements of supernova explosions of a type with computable intrinsic brightness in increasingly distant galaxies, and the exact pattern seen in their apparent redshifts vs. apparent brightness. It was a shocking discovery when it came out, with two separate teams announcing the result pretty much neck and neck. There's also independent and compatible evidence for acceleration from the exact pattern of variations in the temperature of the cosmic microwave background seen at different points in the sky.
The conclusion that the expansion is accelerating was a quite recent result: 1990s, I believe. It's based on careful measurements of supernova explosions of a type with computable intrinsic brightness in increasingly distant galaxies, and the exact pattern seen in their apparent redshifts vs. apparent brightness. It was a shocking discovery when it came out, with two separate teams announcing the result pretty much neck and neck. There's also independent and compatible evidence for acceleration from the exact pattern of variations in the temperature of the cosmic microwave background seen at different points in the sky.