From the Theory Book:
A Do-er would build a business by the sweat of his brow.
A Be-er would build a business by attention to the needs of his clients.
Obviously both Approaches are important, but Main Characters, just like the real people they represent, will have a preference.
A martial artist might choose to avoid conflict first as a Be-er character, yet be capable of beating the tar out of an opponent if avoiding conflict proved impossible.
Similarly, a schoolteacher might stress exercises and homework as a Do-er character, yet open his heart to a student who needs moral support.
When creating your Main Character, you may want someone who acts first and asks questions later, or you may prefer someone who avoids physical conflict if possible, then lays waste the opponent if they won’t compromise.
A Do-er deals in competition, a Be-er in collaboration.
This explanation helps me the most. When I think of a Be-er, I think of someone who would try to plan/think ahead, deliberate over his decisions, put himself in other people's shoes, pretend to be someone else, use his charm, rhetoric or reasoning (appeal to emotion, appeal to reason, appeal to...) over other methods. An Action story might not give him enough time or space to do those things, but he would still try to do them. A Be-er (to me) is more likely to try to convince/coerce/persuade than harm/kill. Internal/Mental rather than External/Physical.
And it seems to me, that when we talk about changing the environment (Do-er) vs. changing ourselves (Be-er), the second option doesn't necessarily mean that we accept how things are. This reminds me of the Dramatica Comic where the Be-er example changes the environment simply by "being" an example. It's more of a direct (Do-er) vs. indirect (Be-er) change of the environment.