This summary is not the rule. The rule text controls.
Both sides in a lawsuit can gather information from each other in whatever order works best for them. You don't have to wait your turn. For example, one side doesn't have to finish taking depositions before the other side can send written questions or ask for documents.
If someone is already gathering evidence, that doesn't stop the other side from doing their own information gathering at the same time. Everyone can move forward with discovery at their own pace, using whatever methods they choose, without waiting on each other.
A judge can change this if needed. If following the normal order would be unfair to someone, or if it would be more practical for witnesses or the parties involved to do things in a specific sequence, the judge can step in and set a different order for discovery. But unless a judge makes that call, both sides are free to gather evidence in any order and at the same time.
