Excellent question that has flummoxed many. To unlock this little mystery let us look at a few other cases where the situation is clearer.
In 2 Sam 24:1 we read: "the anger of the LORD burned against Israel, and he incited David against them, saying, "Go and take a census of Israel and Judah."
But in 1 Chron 20:1 we read: "Then Satan rose up against Israel and incited David to take a census of Israel."
These are both inspired passages but, to us, they both cannot be correct: either God tempted David or Satan tempted David - who was it? Remarkably, to the Hebrew mind, they are both correct because God is omnipotent, nothing happens without Him at least allowing it to happen. So, while Satan did the tempting of David, God is said to have "caused" the problem because He did not prevent it.
In places like Ex 9:12, 10:20, 27, 11:10, etc, we read that the LORD hardened Pharaoh's heart to prevent the Israelites leaving Egypt.
Yet in places like Ex 8:15, 32, 9:34, etc, we read that it was Pharaoh that hardened his own heart.
Again, they cannot be both correct except to the Hebrew mind that God is responsible for what He does not prevent.
In several places we read that God(!!) sends an evil spirit on Saul to tempt him (1 Sam 16:14, 16, 18:10, 19:9). Clearly, God does not have any evil spirits to send. However, again, to the Hebrew way of thinking, God is responsible for that which He does no prevent.
There is a similar thing is Judges 9:23 (evil spirit from God); 1 Kings 2:22, 23, 2 Chron 18:21, 22 all have a "lying spirit" from the LORD.
This idea is technically called the "Divine Passive" where God is held responsible (because He is omnipotent) for that which He does not prevent.
I presume that Eze 14:9 is another example of the same thing. Thus, God does not tempt anyone (James 1:13) but God is held responsible for that which he allows. Therefore, it was the great deceiver, the devil, who did the deceiving.
Ellicott has a useful summary of all this in his inimitable cogent style:
And if the prophet be deceived.—The exact sense of the original is,
“If a prophet be persuaded and speak a word, I the LORD have persuaded
that prophet.” The thought is thus in close connection with what
precedes; in Ezekiel 14:3-4; Ezekiel 14:7, the Lord has refused to
allow an answer through the prophet to the hypocritical enquirer; but
if the prophet, by giving the desired answer, allows himself to become
a partaker of the sin which God abhors, then God will treat him
according to that general method of dealing with sin which is here
described. He “persuades” the prophet in the same sense in which He
hardened Pharaoh’s heart, by making such persuasion the natural
consequence of the immutable moral laws which He has ordained. Men are
held back from sin only by God’s own Holy Spirit drawing them towards
Himself. When they set this aside by transgressing God’s commands, the
inevitable tendency—the tendency under the moral laws God has
established—is to further sin. Hence the prophet who allowed himself
to be persuaded, contrary to God’s command, to answer the hypocritical
enquirer at all, would inevitably be persuaded further to answer him
according to his desires. God does not force men either to receive the
truth or to act righteously. If, notwithstanding His remonstrances,
their hearts are set upon wrong, He will even give them up and “send
them strong delusion that they should believe a lie” (2Thessalonians