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 accept it. Lawyers can't turn down these assignments just because they don't feel like taking the case or would rather work on something else. Courts rely on lawyers being willing to step in when needed, including for clients who can't afford to pay or cases nobody else wants to handle.
That said, a lawyer can say no if there's a good reason. This includes situations where taking the case would force the lawyer to break ethics rules or other laws. It also covers cases where the cost of representing the client would create a serious financial hardship for the lawyer, not just an inconvenience.
A lawyer can also decline if the client or the case is so troubling to the lawyer 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's situation that it would damage their ability to work together or represent the client fairly, that counts as a valid reason to step back. The point of this rule is to make sure people who need legal help can get it, while still protecting lawyers from being forced into impossible or unethical situations.