How is this any different from intrinsic/extrinsic distinctions?
First of all, let me make it clear that I am not unaware of this distinction being the dominant way of distinguishing and categorizing rewards, and that it is passionately promoted by people with a lot more authority than I have. That being said, I wish to borrow your collective ears for a moment to argue the merits of intrinsic/instrumental distinctions.
When examining a reward using the intrinsic/extrinsic model (IEM), we are describing a different set of reward attributes than the ones I think are the most predictive of behavior. IEM describes the mechanics of reward delivery. In this popular model, an extrinsic reward is something somewhat tangible given to the player, whereas an intrinsic reward is something merely perceived. Where an intrinsic reward (in this terminology) might be "feeling satisfied", an extrinsic reward could be a point, a badge or some in-game currency.
In an almost libelously simplified way, IEM deals with who gives the player the reward, where the reward is coming from, and if the reward is tangible/visible to others. Intrinsic rewards are those rewards that the player affords herself ("I will take pride in this") and extrinsic rewards are those that the designer/game gives to the player ("Here's something for your trouble").
So what is the distinction you're making?
In the intrinsic/instrumental model (IIM), the attributes being described are less mechanical, and have more to do with why the reward is rewarding. IIM deals with the player subjective value of a reward, rather than its origin. The distinction is borrowed from value theory (philosophy), and is meant to describe why and how we perceive things as valuable. More importantly, IIM resonates rather well with a lot of data from behavioral economics and behavioral psychology, even though the distinction is rarely made by the people conducting the research. I argue that there is a compelling case to be made that, even at the level of the brain, we respond differently to intrinsic and instrumental rewards.
In IIM, the word intrinsic has a completely different meaning than it does in IEM, the IIM definition being "something that has value in and of itself". An intrinsically valuable thing is a thing that we value because of how we experience it, not because of some use it has. The taste of chocolate is intrinsically valuable, whereas a voucher to buy said chocolate is instrumentally valuable.
This means that something that is extrinsically rewarding in IEM can be intrinsically rewarding in IIM, for example: Being praised by a loved one. IIM considers praise intrinsically rewarding (it just feels good), where IEM considers praise to be extrinsically reward (it required another party to give it to you).
However, interestingly, everything that it is intrinsic in IEM is also intrinsic in IIM, but not vice versa (as demonstrated above).
1) All IEM intrinsic things are IIM intrinsic.
2) Not all IIM intrinsic things are IEM intrinsic.
Neither of these two claims are axiomatic definitions, but are observations we've made.
And you're claiming this helps with predictability of behavior?
Let me first change IEM to internal/external, so we can tell the two models apart. That gives us:
1) All internal rewards are intrinsically valuable.
2) Not all intrinsically valuable things are internal rewards.
Moving on. There's a substantial body of research that our industry likes to cite (Ariely , Glucksberg ) that shows that external rewards can be demotivating and hurt performance. What I find interesting is that there is a very high correlation between demotivation and instrumental rewards.
Praise (external) is rarely (ever?) found to be demotivating, but money can be. In fact, money can even be demotivating by proxy. At this year's GDC Shanghai Dave Mark of Intrinsic Algorithm gave a great talk on the psychology of numbers, where he shared a couple of interesting anecdotes:
A Snickers bar was less motivating when the subject knew its price.
(Here's some enjoyable chocolate vs. the work you are doing is worth 1 dollar)
A bottle of wine was less appreciated when the subject knew its price.
(Here's a token of my appreciation vs. this dinner was worth 20 dollars to me)
Lawyers prefer free work to discounted work.
(I'm doing something because it is the right thing to do vs. I am not being paid what I am worth)
I'll freely admit that I don't recall which studies Dave was refering to (if you know, post it in the comments). I submit to you that external rewards are likely to be motivating when they are intrinsic, and risk being demotivating when they are instrumental.
What does any of this have to do with gamification?
We all like pointing out that the mechanics of a game is not what makes the game, but I submit to you that the mechanics of a reward is not what makes the reward, either. This is not a problem unique to gamification, but something that we often fall short on ourselves, in our own discourse. When we reduce our reward systems into internal/external rewards, we are failing to take the player's subjective experience into account, and miss an opportunity to make valuable predictions.
IIM is a more precise scalpel than IEM, but the two are not mutually exclusive. In fact, once IEM terminology is changed to internal/external we find that there is no conflict between the two models. I hope you'll join me in advocating for this use of the terms, so that we can have clear and productive conversations about design and engagement, and be more credible than gamification proponents.
 Ariely, et al. (2005). "Large Stakes and Big Mistakes". Federal Reserve Bank of Boston
 Glucksberg, S. (1962). "The influence of strength of drive on functional fixedness and perceptual recognition". Journal of Experimental Psychology
 Dave Mark. "Psychology vs. Structure". GDC Shanghai 2013