The Hebrew Scriptures are numbered differently than our English translations and include the title as verse 1 of the psalm [that being said the numbering was added in the early 16th century.]
Most modern Bibles use a different script/font for the superscriptions
Many modern versions add their own titles over the superscriptions
The simple Hebrew word, לְדָוִ֑ד ( le David ) has been translated a number of different ways and has a number of possible meanings; of David, Davidic,
meaning written by David, written in the name and with the authority of David, written about David, etc.
The superscriptions were most likely not original to the Psalms but were added sometime later.
The Septuagint has many more superscriptions than its Hebrew source.
Arguments in the NT were based upon Davidic authorship
*Jesus - Matthew 22:41–46 [Psalm 110]
*Peter - Acts 2:25–29 [Psalm 16)
*Two psalms are called Davidic even though their titles did not show it in the Hebrew (Heb 4:7 [using Ps 95:7–8]) or even in the Septuagint (Acts 4:25–26 [using Ps 2].
Rob