That was an interesting TED Talk, I enjoyed it. Motivation is a very complex, but fascinating thing to think of. You're asking if those three things he listed are the most important aspects to motivation, but I'm not sure the answer can be straightforward. So let's talk about what motivation is first, before talking about how those three things relate to motivation.
We can think of motivation as simply the urge for an organism to do something. But what determines what things the organism wants to do? Activities are done in the pursuit of value. Value itself is an internal thing; determined by the individual and the context the individual finds himself in. Actually defining "value" is a can of worms we don't need to open here. The point is, when a person finds an activity or object valuable, they will pursue it. I think of it from the perspective of neurobiology, and if you're curious you can look into Samejima et al. (2005), and Kawagoe et al. (1998).
So, let's think of the two types of motivators he listed, extrinsic or intrinsic. In the context of value, an extrinsic motivator is when value is placed on an external object. The motivation is to acquire the reward, and the actual activity is made into an obstacle to obtaining it. With intrinsic motivators, value is placed on the activity itself, so the behavior is to perform that activity. An excellent demonstration of this is in what's called the over-justification effect. Lepper et al. (1973) tested children under different reward situations, and found a significant drop in a coloring activity when a big prize was promised beforehand.
So, to get back to this guys three criteria. Autonomy makes sense right away; when a person can choose their own activity they will choose one they find valuable. Additionally, because they chose it, they will likely value it further. This was shown in a kind of funny/sad study by Brehm (1956), where housewives were allowed to choose an item they would supposedly get for free. Their ratings of the value of each item swung significantly towards the item they chose, when compared to their initial ratings. They were never actually given the item, causing one woman to cry.
Mastery is a little more complex. The amount of effort a person is willing to spend on a task is proportional to the magnitude of the reward. By mastery, Pink is actually talking about progress. Consider those little window file transfer boxes, and how they estimate the total time something will take. They look at the current rate of progress, and use that to estimate the total. People do the same thing. If they are making no progress on a topic, they calculate the amount of effort required to complete it to be monumental. If they are making rapid progress, the amount of effort seems far less. Expectancy-value theory has some interesting things to say on this topic, read Wigfield and Eccles (2000).
Purpose is probably the most esoteric, but it is still perfectly possible to understand. Purpose is a feeling that the activity has value in a larger context. Again, value is an internal thing. It is something we construct. Part of the value we assign to an activity is based on the results that activity will have. Feeling that our actions have absolutely no value in a larger context could destroy the value we place in that activity itself.
I like his three criteria, but there are many ways to understand thing like motivation. Like I said in another thread, these psychological constructs are not absolute. They are just ways of thinking of how we work. There are no laws, and no right way to deconstruct them. These three criteria is just one way to deconstruct motivation.
Brehm, J.W. (1956) Postdecision changes in the desirability of alternatives.
The Journal of Abnormal and Social Psychology, 52(3),384-389. [DOI]
Kawagoe, R., Takikawa, Y. & Hikosaka, O. (1998). Expectation of reward modulates
cognitive signals in the basal ganglia. Nat Neurosci. 1(5), 411-6. [PDF] [DOI]
Lepper, M.R., Greene, D. Nisbett, R.E. (1973) Undermining children's intrinsic interest with extrinsic reward: A test of the "overjustification" hypothesis. Journal of Personality and Social Psychology, 28(1),129-137. [DOI]
Samejima,K., Ueda, Y., Doya, K., Kimura,M. (2005). Representation of action-specific reward values in the Striatum. Science 310(5752), 1337-1340. [DOI]
Wigfield, A.,Eccles, J.S. (2000). Expectancy–Value Theory of Achievement Motivation. Contemporary Educational Psychology, 25(1),68–81. [DOI]