Instead of "Training to do pull ups," our mantra should be "Training to be able to pull ourselves up."
This simply means that we train, not to say "I can do pull ups," but to increase our quality of life and our overall fitness. Yesterday my massage therapist reminded me that the older we get, the more flexibility and strength is needed for mobility. Just because we're getting better at a specific skill like pull ups doesn't necessarily mean we're getting fitter overall. The goal is to increase our ability to pull ourselves up when we need to, not to perform a certain number of exercises.
The bottom line? You can train your arms to pull yourself up in many different ways. A vertical pull up is not strictly required. Easier varieties of pull ups (like jumping pull ups or machine assisted pull ups) are no less pull ups!
Have a fantastic day!