There's a "NASA Facts" document for Space Shuttle that says (page 3):
Range safety cloud ceiling and visibility constraints
– Direct visual observation of the shuttle is required through 8,000 feet. This requirement may be satisfied using optical tracking sites or a forward observer.
– A cloud ceiling of 6,000 feet is permitted for short- duration launch windows if all required range safety instrumentation systems are functioning.
– A cloud ceiling of 4,000 feet is permitted if:
a) The cloud layers between 4,000 and 8,000 feet are not more than 500 feet thick.
b) The vehicle can be seen by the Eastern Range airborne and/or the ground forward observers through 8,000 feet and they can communicate with the flight control officer.
A “Good Sense Rule” in effect for launch states: “Even when launch constraints are not violated, if any other hazardous conditions exist, the launch weather officer will report the threat to the launch director. The launch director may hold at any time based on the instability of the weather.”
There's also some information on visibility for return to launch site abort (RTLS) that's relevant to Shuttle and maybe some Falcon 9 missions, but not to an Atlas V launch. The more-current NASA Facts documents for Falcon 9 and Atlas V aren't as open about judgement calls, just having specific limits on minimum ceilings.
These constraints are clearly around range safety requirements. The constraints seemed to give some discretion for range safety officials to decide whether they had sufficient observational coverage.
For the specific questions, in the context of these Shuttle criteria:
1) What is the nature of of a launch visibility requirement? (visible by what, from where?)
Visible by a range observer, from any of a number of possible locations
2) What is the purpose of a launch visibility requirement?
The visibility is required by for range safety operations
3) Under what conditions can it be waved?
Here, there were several ways of meeting range safety's requirements, so less visibility (though still some) was required "if all required range safety instrumentation systems are functioning". Visible observation on a clear day is pretty reliable; if you don't have that other means of observing the flight need to be reliable.
Spaceflight Now's coverage of InSight includes:
The official weather outlook issued today from the 30th Space Wing calls for an 80 percent chance of violating range safety constraints related to visibility during the launch of the InSight spacecraft aboard a United Launch Alliance Atlas 5 rocket. But Col. Michael Hough, commander of the 30th Space Wing at Vandenberg Air Force Base, said the launch team has other ways of meeting the visibility requirement.
"We have a lot of things that monitor the rocket, and if one of those things should fail and ... we have to have visibility from a safety standpoint, then the Air Force would say, yes, now we have to have visibility," Hough said in a pre-launch briefing with reporters and launch guests Thursday.
Local coverage of InSight also includes this quote:
A number of sensors and telemetry monitor a rocket before and during flight. A failure of a key piece of equipment could make visibility a firm requirement for safety reasons, Col. Michael Hough said.
That implies, but does not state, that in the absence of any failed equipment the visibility requirement is more like a guideline.
So why does range safety need to observe the vehicle? What range safety does (aside from writing lots of studies and filling out lots of paperwork) is deal with the consequences of "our rockets always blow up": sometimes the rockets have to be told to blow up before they arrive at somewhere already occupied. So range safety wants to have multiple assurances that everything is going OK. Visually observing the rocket to see how it and it's track looks is part of that. Here's an image of a tracking radar and axle-mounted camera from Vandenberg that "provides data and range safety for missile launches" (more info):
The radar tracks path, keeping the camera aligned.
There's a nice picture of an Appllo-era observing setup here, through it's not clear the video it took was used for range safety purposes.