This summary is not the rule. The rule text controls.
If someone involved in a lawsuit transfers their interest in the case to another person or business while the case is still going on, the lawsuit doesn't have to stop or restart. The case can keep moving forward with the original party still listed, even though they no longer hold the interest being fought over.
However, if it makes more sense to bring in the new owner of that interest, the court can step in and order that change. Someone would need to file a motion asking the court to add or swap in the new party. That request has to be delivered to the other side the same way court papers are normally served under Rule 25.01.
In short, transferring an interest during a lawsuit doesn't automatically change who's part of the case. It just gives the court the option to update the parties if that's the better path forward.