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A creature with several reach attacks can only make an opportunity attack with an attack for which the conditions to trigger an opportunity attack are met.

The best available authority on this issue is actually Jeremy Crawford's guidance on adjudicating opportunity attacks by a PC with the War Caster feat.

Bear with me here. One of War Caster's benefits is to let you cast a spell instead of making an opportunity attack. See PHB p. 170. Of course, many spells have a range greater than the standard 5 feet. So what happens when a PC with War Caster is wielding a 10-foot-reach weapon, like a polearm? Because of the PC's greater reach with the polearm, an opponent ordinarily would trigger an opportunity attack from the PC if it moved from 10 feet to 15 feet. So, if the PC also has War Caster, could she opt to cast a spell instead? Crawford says no:

Q: If I have the war caster feat and a reach weapon can I use a spell instead of the weapon if they move to 15 ft?

A: The intent is that any OA triggered because you're wielding a polearm is then made with that polearm.

In other words, the PC can't cast a spell instead of making an opportunity attack because the spell isn't what gives the PC reach. The polearm is what gives the PC reach. Thus, the polearm is the only attack method as to which the conditions to trigger an opportunity attack are satisfied by the target's leaving that reach.

Logically, there is no reason to treat any opportunity attack from any creature, whether monster or PC, any differently. If a creature has an attack method that offers a certain reach, it gets to make an opportunity attack with that attack method if and only if the target's movement triggered the opportunity attack at the reach of that attack method.

The adult dragon used as an example in the question has:

  • a claw attack at ordinary 5-foot reach
  • a bite attack at 10-foot reach
  • a tail attack at 15-foot reach

If a PC moves from 5 feet to 10 feet, the only opportunity attack the dragon can make would be with the claw. The PC is not moving out of the dragon's reach for purposes of the bite or tail attacks.

If the PC moves from 10 feet to 15 feet, the only opportunity attack the dragon can make would be with the bite. The PC is not moving out of the dragon's reach for purposes of the claw or tail attacks. (And anyway the PC would be out of reach of the claw.)

If the PC moves beyond 15 feet, the only opportunity attack the dragon can make would be with the tail. The PC is not moving out of the dragon's reach for purposes of the claw or bite attacks. (And anyway the PC would be out of reach of both.)

Finally, just to complete the thought: imagine a creature with several attack methods that all have the same 10-foot reach, but only one that says it can be used to make an opportunity attack when a PC attempts to cast a spell. If a PC attempts to cast within 10 feet of the creature, we would not say the creature can make any kind of opportunity attack it wants just because the PC's action satisfied the conditions for one particular kind of opportunity attack. The opportunity attack would be made with the particular kind of attack that says it can be made under those conditions.

Reach should work no differently. It provides a set of conditions for making an opportunity attack. If an opponent does something to meet those conditions, the creature with reach gets to make the kind of opportunity attack for which the conditions are met, and nothing else.