This summary is not the rule. The rule text controls.
When a court asks a lawyer to take on a case, the lawyer usually has to say yes. Lawyers can't just turn down an appointment because they don't feel like taking the case or don't like the client. Courts need lawyers to step up and represent people who need help, especially when those people can't afford their own attorney.
That said, there are a few good reasons a lawyer can refuse an appointment. A lawyer can say no if taking the case would force them to break ethics rules or other laws. They can also refuse if the case would cost them too much money to handle, more than what's reasonable to expect. Finally, a lawyer can decline if the client or the case is something they find so offensive that it would seriously hurt their ability to do a good job representing that person.
These exceptions exist for real, practical reasons. Nobody benefits from forcing a lawyer to take a case they can't afford to handle or one where their personal feelings would get in the way of doing solid work. But outside of these specific situations, lawyers are expected to accept court appointments as part of their job.