The Jewish Study Bible, in its notes on Psalm 2, defines “royal psalms” as “those concerning kings.” It goes on to say that “None [of them] mentions a specific king by name, and their origin and uses remain obscure.” It also gives a “possible list of royal psalms” which is reproduced below. To the original list I have added italicized elements. The first to be noticed is that I added the verse citations for any explicit reference to the “king” that appears in the psalm. Some of the psalms do not make explicit reference to the king as “king,” and, because of this, do not show a verse citation.
Psalm 21:1, 7
Psalm 45:1, 5, 11, 13, 14, 15
*These verses were not on the original list but are included here because they are psalms that include an explicit reference to a “king” who is not God. I did not include psalms where merely a synonym or allusion to the king appeared (such as Psalm 28, where a reference to “His anointed” occurs in verse 8).
The original list is from The Jewish Study Bible, ed. by Adele Berlin and Marc Zvi Brettler, Oxford University Press (2004) p. 1285
Herman Gunkel is a notable scholar of the Psalms. For more on his classification (upon which the Jewish Study Bible‘s list would seem to be based), see this post.