Looking at the list of German infantry divisions they all have the Motorized or Panzergrenadier added to their name to differentiate them from foot divisions.
The American and British infantry divisions are all uniform in their naming conventions. So where their armies completely mechanized or did mechanization not impact their naming conventions?
The difference between a German motorized division and a foot infantry division was twofold. One, the motorized division possessed organic motor transport (trucks) for troops; in foot divisions, the men walked wherever they went. Two, the motorized division's trains were fully motorized, whereas foot divisions were heavily reliant on horses to haul artillery, baggage, and supplies. Panzergrenadier divisions are a bit different in that they generally included a battalion of tanks or assault guns.
In US divisions, the trains were fully motorized; US formations did not rely on horses outside of a few scenarios where the terrain was too bad for anything except impromptu mule teams. However, the leg divisions lacked organic motor transport - that is, the infantry had to walk. There were early attempts at organizing motorized divisions - 4th ID was motorized for a time - but these were done away with by early 1944. The US Army generally had enough trucks available as corps assets that infantry divisions could be temporarily motorized as needed, and it wasn't necessary to permanently dedicate trucks to moving individual divisions.