Your reasoning is flawed, but your result is correct.
Technically, combat only has one type of check. a Combat Check. Spells and weapons allow the user to choose which die to use other than the default Strength die, and also add Traits that allow additional affects to apply.
Shortsword: "for your COMBAT CHECK, reveal this card to roll you Strength or Melee die +1d6..."
Shortbow: "for your COMBAT CHECK, reveal this card to roll your Dexterity or Ranged die + 1d6..."
Acid Arrow: "For your COMBAT CHECK, discard this card to roll your Arcane die +2d4..."
Note that none of these cards say anything about changing the check from a combat check to a "Melee/Ranged/Arcane" combat check. They are all combat checks, they simply allow the user to choose different dice to roll for the check.
A combat check using a Shortsword has the traits: Sword, Melee, Piercing, Finesse, Basic.
A combat check using a Shortbow has the traits: Bow, Ranged, Piercing, 2-handed, Basic.
A combat check using an Acid Arrow has the traits: Magic, Arcane, Attack, Acid, Elite.
Thus, other cards which relate to those traits can have an effect. A shortsword has the piercing trait, so those monsters with a resistance to piercing would gain their resistance. A shortbow has the ranged trait, and thus cards like the archer can be used to affect it. Acid arrow has the magic trait, and may thus affect monsters that are only affected by attacks with the magic trait. Etc.
Note again that there is actually no "Ranged Combat Check". There is only a Combat Check with the ranged trait. Therefore, it appears that the intent of the wording "Ranged Combat Check" and "Melee combat check", etc, are actually "Combat check with the Ranged/Melee/Arcane/Etc trait".
It is obviously faster to write/say "Ranged Combat Check", than "Combat Check with the Ranged trait." It is probably just a common shorthand they used that was clear and obvious to those testing, but became less clear when shown to a broader audience who weren't intimately familiar with how the game works.