This summary is not the rule. The rule text controls.
When a judge appoints a lawyer to take a case, the lawyer generally has to accept it. Lawyers cannot dodge these appointments just because they don't want the case or don't like the client. Courts rely on lawyers to step up and represent people who need help, including those who cannot afford their own attorney.
There are only a few valid reasons a lawyer can refuse an appointment. First, if taking the case would force the lawyer to break professional conduct rules or other laws, they can say no. Second, if representing the client would create a financial hardship that's too much to bear, that's also a valid reason to decline.
Finally, a lawyer can refuse if the client or the case itself is so objectionable to them personally that it would likely damage their ability to do a good job representing that client. This isn't about simple preferences or minor annoyances. It has to be serious enough that it would actually get in the way of a proper attorney-client relationship. Outside of these specific situations, lawyers are expected to take on appointed cases as part of their duty to the legal system.