Your latter definition is more formally known as alpha. That's the ability to outperform an index on a risk-adjusted basis. Discipline is a good trait, but you can be disciplined in following the worst trading system ever. Discipline alone will never make anyone profitable.Depends on definitions.
Some people define edge as anything that gives you a competitive advantage in the market. In this case, discipline is an edge.
Others would say that edge refers specifically to informational advantage. In this view, it is your trading system that contains the edge. Discipline alone is not a trading system.
Alpha in Investing: Definition, Formula, Uses (businessinsider.com)