This answer was correct as of the question's asking. As later revisions to the UA Mystic came along, they invalidated this answer. This answer contains the latest official description of psionics.
In the D&D podcast of July 10 Mike Mearls describes it, starting at about 14:00, as
The power of the mind. It is a power source that comes from within... it derives from the opening of the mind's eye....
He then goes on to discuss how the Far Realm in D&D cosmology "bumps into" our planes and "can twist reality a little bit." Continuing,
What happens is, if you're psionic your mind's eye is open and you see the underlaying structure of this reality in a way that other people don't. And by drawing on your willpower and your own mental energy you can exert changes to the structure of reality around you- or you can perceive things.... Or you can understand your body and manipulate your muscles beyond what would normally be possible.
He goes on to stress the separate nature of psionics and magic, describing how they've intentionally renamed many psionic abilities to avoid spell-names.
As for the distinction with ki, the class description of a monk specifies that monks have made a specialized study of "an element of the magic that suffuses the multiverse--specifically, the element that flows through living bodies." (PHB p.76)
So monks are using one form of the magical energy that's native to the multiverse; mystics are in tune with things coming from farther out.
Related reading (from comments):
Psionics (role-playing games) - Wikipedia
Why do psionics suck? - AngryGM
As Unearthed Arcana rules, this writing is a bit unpolished. Here's how I understand it:
You spend your action and gain the psi points; that's all you get from that action (the rule says you gain points that you "can" spend, not that you gain points and spend them). Those psi points are tracked separately from your normal pool.
Whenever you use a discipline later to create an effect, you choose whether to spend your normal psi points or your special psi points on that effect. You can't mix and match for a single use of an effect, but I see no rule that prohibits spending special psi points on one effect and then spending normal psi points on an effect from the same discipline.
If one were to interpret the rules to say that using special points locked a discipline from being available via normal points, there would need to be some rule that explained when that lock expired. Since there isn't such a rule, the rule against mixing point types must only apply to a single use of an effect.
Best Answer
The Psionic Point limit is a hard cap for a single expenditure
No, you "can't do this all at once" since nothing costs more than 7 points. Since you need to still adhere to the "as an action" or "as a bonus action" limit, the standard action economy applies.1
None of the specific wording in the Psionic Mastery description offers an exception to the hard cap on how many points you can expend to activate a psionic discipline (one instance of a psionic discipline). You will note that none of them costs more than 7 points, so you'll typically be able to do two things, at least, from that bonus point lump this ability offers.
You'd need to use some of the points on one action, and the rest either on another action, or a bonus action. Once you've activated this, those points are yours to spend until you finish a long rest.
As written on page 4 of the UA document (Psi Limit):
That's not a problem, since the Psionic Mastery points can be used for more than one thing, and you can stack various psionic abilities that are eligible which is not something you can do with your regular Psionic Point budget.
Example 1:
You use an action to establish this bonus pool of points, 9, and you then use (and stack) two different instances of a discipline (Telepathic Contact (p. 26)) Occluded Mind (2 points) and Psychic Domination (7 points). That uses up all of your points, and each of them cost 7 points or less. Your psi limit topped out at 9th level, at 7, and you didn't not earn Psionic Master (1/day) until 11th level.
Example 2:
You spend 2 points on Occluded Mind (as above) on one opponent, and the other 7 points on a different area near you on Psychic Crush (P. 24) to damage a small crowd of enemies.
Example 3:
You spend 2 points on Occluded Mind (as above) and as a bonus action spend 7 points on a 5th level spell slot (Arcane Dabbler, p. 8) that you will retain "until you use it or finish a long rest."
1 Ruling territory here, since on page 3. it states that "psionics is a special form of magic distinct from spell casting" but it doesn't say that it has a different action economy. The only 'special' feature is the ability to stack concentration effects when using this particular feature of the class).