I think the issue is that most of the manufacturer volume ratings on skimmers are artificially high because they assume almost no bioload. If you check out Bulk Reef Supply's website, they give their own volume rating for each skimmer. So for example, my Reef Octopus 110 INT skimmer is manufacturer rated for up to 100 gallons. Bulk Reef Supply however only recommends it for tanks up to 75 gallon. I have it running on my 50 gallon system, which is heavily stocked and I definitely would not want a smaller skimmer. Mine is definitely not oversized. Marine Depot also has their own volume recommendations for skimmers based on whether your tank is lightly, moderately, or heavily stocked. They rate the skimmer I have for only 60 gallons if heavily stocked.
Since you're moderately stocked, you may not need a skimmer that is rated for quite twice your volume, but you definitely will want one that's rated for larger than your total water volume. I'd guess like 50% over rated would be good. So since you have 120 gallons, I'd say you'd want a skimmer rated for 180 gallons at minimum.