This summary is not the rule. The rule text controls.
When a court asks a lawyer to take on a case, the lawyer generally has to say yes. Lawyers can't turn down these appointments just because they don't want the case or find it inconvenient. Courts rely on lawyers to step up when needed, especially for people who can't afford their own attorney.
That said, there are a few good reasons a lawyer can refuse the appointment. First, if taking the case would force the lawyer to break ethics rules or other laws, they can say no. Second, if representing the client would cost the lawyer too much money and create a real financial hardship, that's also a valid reason to decline.
Finally, a lawyer can turn down a case if the client or the situation is so troubling to them personally that it would get in the way of doing a good job. For example, if a lawyer has such strong personal objections to a client or their case that it would damage their ability to work together or represent them fairly, they don't have to take it. Outside of these specific situations, though, lawyers are expected to accept court appointments as part of their professional duty.