The CDC defines "moderate-intensity" as "brisk walking". Other than that, they are very vague on what constitutes "light", "moderate", and "vigorous" because that is going to be different for the individual. Running up the stairs for a person just getting started will probably be vigorous, but it's just a Tuesday for an Olympic athlete. What generally matters is what your heart rate is while you're performing the work.
Moderate exercise is usually at an intensity you can sustain for a very long time; basically your body will be forced to stop before your energy levels do. Vigorous workouts are sustainable for 10-30 minutes before you gas out. In the running communities, a common way runners gauged the intensity of their training was based on how many words they can speak without taking a few breaths. If you can hold a conversation normally, then you are within that "moderate" cardio range. If you need a break every few words to take a few breaths, then you're approaching that "vigorous" range. If you're gasping for air almost every two words, then you're close to your maximum ability. Since runners are usually alone, they sing their national anthem or something.
In the case of walking up stairs, it's certainly more intense than brisk walking, but I don't think it's intense enough to bring someone in that vigorous range. Running up the stairs will certainly bring you in that range and fast (although use caution because there's the added danger of falling).