is of a different type. He and the other Istari wouldn't need physical bodies
if they worked through supernatural means. In fact, the Valar could have
dispensed with the middle men and just done it themselves in that case. But,
then, no one would be able to make any free choices.
Gandalf teaches, leads, persuades, threatens, and lends actual physical aid, in
the form of a wise old man. A poweful old man, but one who is forbidden from
using all of his power. That's significant, I think.
Working in this way allows the others to distinguish between Gandalf and
themselves, to decide for themselves whether to accept his advice and council
or not.
On Amon Hen, Frodo can't tell the difference between his thought and Sauron's,
but he is aware of Gandalf's voice unambiguously.
This also gets back to the whole point of the brooch dropping. Do we really
believe that Aragorn would have given up the hunt if he hadn't found it? That
he would have stopped 5 leagues from the battle ground, not gone on to examine
the ground for himself?
So, if a higher power is involved, what was the point in manipulating
Pippin?