I am confused by the way you add additional damage for extra successes in a combat check. Upon reading the normal personal combat check, it seems to indicate that every uncanceled Success grants +1 damage. So if you rolled 1 fail and 3 successes, you'd get +2 on the combat check. Here is the exact text:

When making a combat check, if the check is successful each uncanceled * adds + 1 damage to a
successful attack.

However, in the vehicle/ship combat check it seems to indicate that 1 "success" is spent to "win" combat and only *additional* uncanceled successes beyond the first grant a +1 bonus to damage. Here is the text from that section:

When making a combat check, after calculating overall success, each remaining * adds + 1 damage to a
successful attack.

Is this the correct interpretation? It seems odd that the rules would be inconsistent between the two types of checks.

It's poor wording, but not supposed to be inconsistent. Damage is always base damage plus net successes.

Yeah, in the ships chapter example, basically the correct interpretation is this: "after you get rid of any success cancelled by failure, and you still have success symbols showing, each remaining success adds +1 damage to the attack."

So in short, it is consistent.