The short answer is no, there is no specific requirement for VFR flights to state an emergency, however you may be expected to explain yourself and your actions once you reach the ground.
My assumption is no because a 2-way radio is not even required in all
To expand on this you don't need a radio on board for VFR flight (FAA FAR 91.205) so they would have a legally contradicting issue if they required you to announce when you don't have a radio. As mentioned by Ron in the question you linked the rule is "Aviate, Navigate, Communicate" If you are too busy with the first two that is far more important.
Im not sure where they are pulling the quotes but this NASA presentation on emergencies sums it up well.
“…an intent of 91.3 is to ensure the PIC will handle the emergency in
a manner necessary to save lives and not be worried about regulatory
“I’ve never seen a pilot violated for deviating from a
regulation when that pilot has either declared an emergency OR has
stipulated in ANY written response to the FAA that an emergency
existed at the time of the deviation.”
This article has some interesting points from a lawyers perspective on the situation,
Madsen defended a case for a pilot who deviated from an assigned
altitude due to extreme turbulence while flying a Cessna 150. The
pilot had dropped his handheld mic, so he couldn't initially
communicate his situation. He stabilized the airplane and was about
600 feet off his altitude when ATC called him. At that point he was
able to find the mic and he told the controller what had happened.
"It's unrealistic in some situations to communicate your emergency
before you secure the aircraft," Madsen said. "You want to control the
aircraft first." Madsen said the case was ultimately dismissed, but he
felt it would have been much easier to defend had the pilot declared
Here we can see that ultimately the pilot was in the right and while he should have declared, there was no legal grounds that required it.
On some level you are at the mercy of the FAA with this one and its clearly a case by case basis (as is everything in aviation) and the lawyer in the above article goes on to say
Some FAA field offices take the wording very seriously. Such was the
case with the Cessna 150 pilot who deviated from his altitude. "They
questioned why he didn't state an emergency and call out mayday, and
they took great issue with the fact that he didn't use the accepted
terms," Madsen said.
Again the case was ultimately dismissed whether or not they were happy about that.