When hitting different parts of a monster, the damage numbers show up in either grey or orange. What is the difference?
From a first look, it seems like the orange numbers indicate a monster's weakspots. If so, what hitzone value is required for orange numbers?
Yellow/orange damage numbers indicate that you are hitting a Monster's weak-point and white/grey numbers are anywhere that is not a weak-point.
A Monster's weak-points are detailed in the Hunter's Guide Monster Manual once you reach level 1 research for that Monster. Here is the Monster Manual page for the Great Jagras. The 3 icons (slashing, blunt, and ranged) indicate the type of damage that weak-point is weak too, more stars equals more damage.
There are also indicators for affinity (positive and negative crits), which are small yellow stars on damage numbers for positive affinity and blue stars for negative affinity.
You're right in your assumption that orange and yellow numbers show up when you hit a monster's weak points. These are nice visual indicator of whether or not the positioning of your attacks is right - if you're seeing grey numbers, aim somewhere else.
I'm not sure exactly what the bounds are for the hitzones however - more research is probably necessary for the community to figure out what the 'translation' is from hitzone to damage colours.
The colour of damage numbers actually depends on both weapon sharpness and hitzone value of the part being hit, not just the latter.
For ranged weapons, which do not use sharpness, orange numbers appear when hitting hitzones of at least 45.
For melee weapons, sharpness makes it possible to see orange numbers on hitzones of at least 40. These hitzones are not shown as 3* weaknesses in the monster guide, and do not trigger Weakness Exploit. This is easy to see in the training area, on the small, white barrels.
The colour of damage numbers is also not affected by the elemental hitzone of the part being hit, except in the case of fully elemental attacks like the Charge Blade's element phials.
