As an introvert, I can be a social butterfly at parties but need to leave after an hour because it felt like work. As an extrovert, I can be a wallower but want to stay out all night because the ambiance is energizing.

No, that definition you linked to does not imply low social skills.

Social skills are the ability to make conversation, to make people feel comfortable, to engage others, etc.

Extroversion is the desire to do those things.

Yes, they are clearly correlated, but by absolutely no means interchangeable. There are plenty of introverts who have perfectly decent social skills, and plenty of extroverts who have terrible social skills, because the desire for something is not the same as having a skill for it. For example, an adolescent extrovert may feel strong motivation to be outgoing, loud, talkative, and active, but that doesn't mean he will have good social skills and be polite.

The most common psychological definition of extrovert and introvert that we currently use was largely shaped by Carl Jung, and his definition involves far more than just sociability. However, sociability is the easiest application. What it really comes down to (as the link you provided points out) is a focus on the "external world" vs the "internal world", so I use the word "are" in those two definitions above very loosely.

Introvert has a definition which is colloquial, which is what you're pointing to.

It also has a definition used in psychology as well, which is something completely different, than the term used colloquially.

Enjoying socializing (extraversion) != being good at it (social skills)