As you stated, there are two types of endurance - Cardiovascular, which is the ability of the body to deliver blood, oxygen and nutrients to working muscles, and the ability of the muscles to use that oxygen to do work. The other is muscular endurance, which is the ability of a muscle to do work without fatiguing.
Cardiovascular endurance is stressed by doing exercise that elevates the heart rate and respiratory rate for extended periods of time. As you are first starting out, you may feel that you could run or bike (or whatever) much further, but that you can't breathe. As you keep stressing the CV system, it will adapt by being able to pump blood more efficiently, make better use of the oxygen being delivered, and other physical adaptations such as growing new capillaries into the muscles being used.
Muscular endurance has a strength component to it as well, which also plays a part in the endurance factor. The more you train for strength, the less endurance you will have at that workload. The more you train for endurance, the less strength. They counterbalance, so while someone might only be able to lift 500 lbs once, they might be able to lift 100 lbs for quite a long time.
Where your instructor is coming from, is he is stating that people have the grip and foot strength to hold on to a technical hold, but that the ability to do so for a long time is lacking. This is what needs to be trained, and the way to do that is simply to do more of the activity, in a progressive loading fashion.
You have probably noted it yourself, in that when you first started rock climbing, your forearm muscles grew fatigued on simple climbs, after a certain period of time. As you become more efficient and more trained, the simple climbs don't tire you out, and now it's the intermediate climbs that pose the challenge. As you keep progressing, it builds upon itself.
The difference in training between pure strength and pure endurance, and being able to mix them effectively for the activity in question is why many trainers now have 4 year degrees and other advanced training, as there are many different complex systems being used, as well as being able to prescribe specific types of training for specific sports.