I would propose this analogy:
Psionics is to Magic as Cybernetics is to Technology.
You could, for example, have a blade embedded in your cyber-arm. It will always be with you, and can't be taken away. It's good at its job, and doesn't occupy your hands while being used. However it is only ever what it is, barring an expensive and complicated re-fit.
On the other hand, you could have a blade that you hold in your hand. You need some place to store it, and it can be taken away from you. But you have the option for anything from a vegetable peeler to a filet knife to a butcher knife to a knife-on-a-stick (ie: polearm) to Cloud's airplane wing of a sword.
Magic and Technology can be anything you can figure out how to make, and are incredibly versatile. But at the same time, they require that you "do stuff" to use them — hold the sword in your hands, or the binoculars up to your face, or wave a wand around while you perform a chant.
Cybernetics and Psionics, on the other hand, don't need that 'manual' work performed in order to gain the benefits of their functionality. You hit a mental switch, and the thing is done, or available for use. However, for all their compact efficiency, the scope of what they can do is much narrower than their external counterparts. Likewise, there's a limit to how much you can "stuff" inside your body/head.
My assessment, then, of whether psionics is magic, is "Yes". It's a more efficient form of magic, both in storage and execution, but it is, in the end, magic.
An EMP that shuts down the phone in your hand will also shut down the phone in your head. An anti-magic field that shuts down a standard magical effect would also shut down a psionic magical effect. Dispel Magic would work the same either way, because it's affecting the magical effect, not the means of producing that effect. Concentration would also work the same way.
There are really only two sticking points, and both depend on the ability to detect the execution of the spell. The first is Counterspell (ie: if you can't see the spell being cast, you can't counter it), and the second is detecting who cast a spell. I would be willing to give up the option to Counterspell if it was required that use of psionics always leaves a visible sign — glowing eyes, flames licking over the body, whatever. Thus it would never be possible to be truly anonymous with the use of psionics (eg: Charm Person at the royal ball), even if other spellcasters are not able to explicitly stop you (a side benefit of the efficiency gained by giving up flexibility).