What is definition of weight of person in elevator moving with acceleration?
The ISO agrees with you. Their definition of weight is $mg$, where $m$ is an object's mass and $g$ is the local acceleration of free fall. Note that this is more general than the second definition that you quote, which is earth-centric. Using the ISO definition, an object's weight in an accelerating lift does not change, but an object's weight on the moon is different from its weight on earth.
The reaction force on an object (as measured by a spring scale for example) is sometimes called apparent weight. So apparent weight does change in an accelerating lift.
The ISO definition of weight excludes the effect of buoyancy (otherwise a floating object would have zero weight - and a submerged object that was less dense than water would have negative weight !). However, it rather oddly includes the effect of the earth's rotation.
There is a discussion of the various definitions of weight in this Wikipedia article.
It's really vague to ask how much something "weighs" in an accelerating frame because, well, there's no one answer. A better way to phrase the same is "What would a weighing scale read when the object sits on top of it in a given frame?" Now while they seem to ask the same thing, there is a subtle difference. The latter let's us escape from the technicalities of the situation which as a physicist, is in good spirit because we want to keep things flexible and not restrict ourselves to the "definitions" "we" "made".
What is the weight of a person of mass m if the elevator is moving with an acceleration a?
It depends on how you define weight and how flexible you are with your definition. One might say the weight is a constant because it's just mass time g, the acceleration due to gravity: an orthodox physicist. Another might change his definition of "weight" to calculate a more useful quantity, say the minimum strength of the wooden plank base of a lift which accelerates up at a given rate for 3 people to safely reach the upper floors. At the end of the day, it's a fight between usefulness and technicality.