We are a commune of inquiring, skeptical, politically centrist, capitalist, anglophile, traditionalist New England Yankee humans, humanoids, and animals with many interests beyond and above politics. Each of us has had a high-school education (or GED), but all had ADD so didn't pay attention very well, especially the dogs. Each one of us does "try my best to be just like I am," and none of us enjoys working for others, including for Maggie, from whom we receive neither a nickel nor a dime. Freedom from nags, cranks, government, do-gooders, control-freaks and idiots is all that we ask for.
Very few exercises are pure forms of those four categories, but there are differences in emphasis which is why we recommend balanced programs of resistance, calisthenics, and cardio training to maintain or improve fitness.
Strength is the ability to move things which resist moving. Power is the ability to move things (including yourself) with speed and force. For example, bench press and rows are mostly strength exercises. Powerlifts are power exercises: deadlifts, squats, military press, etc. in which bursts of speedy intensity are required.
Where would we categorize pull-ups? I'd say Strength.
We have discussed cardio training at length. The main muscle it trains is the heart muscle. While any difficult exercise stresses the heart, only pure cardio training (HIIT via HIIT calisthenics/ aerobics class or sprinting intervals) gives the heart a specifically strength-building stress.
So what about endurance? If you are somebody who "gets too tired" from non-resistance activities, you have an endurance issue. It is not rare for very strong people to have poor endurance or for high-endurance people to be relatively-weak. We want both strength and endurance. Anything that is high-rep builds endurance but not strength or power: long-slow "cardio", high-rep (10-20) resistance work, calisthenics.
One caveat: Do not ever do high-rep (over 10) deadlift sets. The human body is made for low-rep heavy floor lifting (8 or fewer). If you can do over 8 deads, you need to increase the weight and reduce the reps.
The sport called Olympic weightlifting would be better termed power lifting because the lifts contested there, the snatch and the clean & jerk, require great power to complete successfully. There is no such thing as a slow snatch. A slow clean is not racked at the shoulders.
The sport called power lifting, however, is much more a display of strength than power. A heavy deadlift can take 5-10 seconds to complete. Same with squat, bench press and OHP. A squat you can do quickly is too light to do anything for you.
Of course, improving your deadlift means you'll be able to clean & jerk more, because power is force divided by time, and if you can apply more force, you will automatically be able to produce more power. There are of course limits to this and technique plays a very important role especially in the Olympic lifts.
Good caveat and the end there. Heavy deadlifts are very stressful to the entire body and one heavy set of five is adequate.
A set of 5 heavy deadlifts is HIIT. After 5 heavy deadlifts my heart is pounding, I am sucking air and my legs are wobbly. Same goes for squats. Presses and bench press don't have the same effect. Barbell Logic has some good videos on youtube and some good podcasts regarding strength training.