What actually changes between these three trips
Most ski-trip packing lists are identical regardless of resort. A few things genuinely differ here based on elevation and terrain.
The base list, wherever you're going
Layered thermal base layers, a waterproof and insulated outer shell, ski socks (not cotton), goggles rated for variable light, sun protection at altitude (the UV exposure on snow at 2,000m+ is stronger than it feels), and a helmet — increasingly standard rather than optional at all three resorts.
If you're heading to St. Anton's off-piste terrain
Avalanche safety kit — transceiver, probe, shovel — is not optional if you're leaving marked pistes, and it's rentable locally if you don't own it. If you're booking a guided off-piste day, ask the guide office in advance whether the kit is included or needs to be arranged separately.
If you're splitting time in Innsbruck the city
Because Innsbruck's ski areas are day trips out from a genuine city base, it's worth packing at least one non-ski outfit — evenings in the old town don't call for ski gear, unlike a purpose-built resort village where everyone's dressed for the mountain all day.