Who incited David's census
Yahweh incites David to number Israel in 2 Samuel; Satan incites him in 1 Chronicles — the same episode with two agents in the parallel histories.
2 Samuel 24:1-2
1Again Yahweh’s anger burned against Israel, and he moved David against them, saying, “Go, count Israel and Judah.”
2The king said to Joab the captain of the army, who was with him, “Now go back and forth through all the tribes of Israel, from Dan even to Beersheba, and count the people, that I may know the sum of the people.”
1 Chronicles 21:1
1Satan stood up against Israel, and moved David to take a census of Israel.
What each tradition says
Rabbinic / Midrashic
The Jewish Encyclopedia notes that in both passages Satan is merely an accuser acting only by God’s permission. Yet in 1 Chronicles 21:1 Satan appears able to provoke David to destroy Israel. The Chronicler, it says, regards Satan as an independent agent, especially striking because 2 Samuel 24:1 says God Himself moved David.
The Jewish Encyclopedia (New York: Funk & Wagnalls, 1901-1906), s.v. 'Satan', The Jewish Encyclopedia, s.v. 'Satan' — section 'In the Bible' (on 1 Chronicles 21:1 and 2 Samuel 24:1)
Patristic
Origen says that anyone who reads in 2 Kings of God's "wrath" inducing David to number the people, and in 1 Chronicles that the devil suggested it, will see by comparing them why the "wrath" is mentioned. That wrath, he says, is the wrath of which Paul declares all are by nature children.
Origen (c. 185-c. 254), Against Celsus, Book IV (trans. Frederick Crombie), in Ante-Nicene Fathers, vol. 4 (Buffalo: Christian Literature Publishing Co., 1885) — Against Celsus, Book IV, ch. 72 (on the 'wrath' of God and David's numbering)
Catholic
The Catholic Encyclopedia states that in 2 Samuel 24 the sacred writer records a sin of pride on David’s part in his resolution to take a census. As penance, David was allowed to choose famine, an unsuccessful war, or pestilence; he chose the third, and in three days 70,000 died.
The Catholic Encyclopedia, s.v. 'King David' (New York: Robert Appleton Company, 1907-1913), The Catholic Encyclopedia, s.v. 'King David' — the paragraph on 2 Samuel 24 (David's census and its penance)
Orthodox
No sourced position found. Checked John of Damascus (Exposition), Gregory of Nyssa (Against Eunomius, Answer to Eunomius, On the Making of Man), the 1672 Synod of Jerusalem decrees, and Chrysostom's homilies on Romans/John for 'number/numbered Israel', 'Satan' with David, and the census: no Orthodox statement found on the two accounts.
Reformed / Protestant
Matthew Henry holds that God is not the author of sin and tempts no one. Satan provoked David to number Israel, suggesting it as a sin, as he put it into Judas’s heart to betray Christ. God, as righteous Judge, permitted it, using David’s sin to punish Israel for other sins, which he might justly have punished without it.
Matthew Henry (1662-1714), Commentary on the Whole Bible, vol. 2: Joshua to Esther (1706-1721) — Exposition of 2 Samuel 24:1 ('the anger of the Lord was kindled... he moved David')
Evangelical
Wesley explains that Satan stood before the Lord and his tribunal to accuse David and Israel and to beg God's permission to tempt David. He notes that standing is an accuser's posture in human courts, and Scripture speaks of divine things in human terms, elsewhere depicting Satan in this posture.
John Wesley (1703-1791), Explanatory Notes upon the Old Testament (1765; CCEL's 'Wesley's Notes on the Bible') — Note on 1 Chronicles 21:1 ('And Satan stood up against Israel')