It is extremely common for extensions to be agreed upon between lawyers as a "courtesy." This is particularly true when we are talking about the defendant filing an answer.
As a technical matter, whenever a court deadline is to be extended, a motion should be filed with the court. This is true even if both sides agree. That said, if both sides agree, and no hearing is set, it is very common in Florida STATE courts for the parties to simply agree to the extension without filing anything with the court. This is uncommon in Florida FEDERAL courts, because judges get reminders from the clerk about deadlines.
If an agreement was reached to extend the time to answer, I would normally not advise that a motion for default be filed. The clerk of court might automatically enter a default, but the defendant will likely be able to get the court to set aside the default. The plaintiff will then look less credible to the court, because the plaintiff told the defendant one thing and then told the court another.
If an agreement was not reached, but a motion was served and not filed, the plaintiff may be best advised to file the unusual motion for court's default (as opposed to a motion for clerk's default). A clerk's default may not be appropriate in this circumstance, because a clerk's default can only be entered if the defendant has failed to file OR serve any paper in the action. A court's default is proper when the defendant has failed to file an answer, file a motion to dismiss, or other present defenses.
Be aware of the fact that the court or clerk must enter a "default" prior to the court entering a "default final judgment." Typically, when a defendant does not file or serve anything, a plaintiff files a "Motion for Clerk's Default," waits for the clerk to enter a default, and then files a "Motion for Default Final Judgment" (or, if the judge is open to it, just sends a letter to the judge with a copy of a proposed final judgment).
In your case, you might follow this same procedure, seeking a "Court's Default" rather than a "Clerk's Default," but it would not be uncommon to do a single "Motion for Court's Default and Default Final Judgment." There are advantages and disadvantages to filing the two motions as a single joint motion. I recommend speaking further with an attorney to determine which procedure is appropriate in your case.
Best of luck to you.
The information provided in this and other answers on Avvo are general in nature and limited to the facts as stated. The information provided in this and other answers on Avvo should not be construed as legal advice on which the reader relies without further consultation with an attorney. No attorney-client relationship is created on Avvo question & answer forums. This attorney is licensed and admitted to practice law in the State of Florida only.