Our Rabbis taught: As for all the songs and praises to which David gave utterance in the Book of
Psalms etc. 'To David, a psalm’ [לדוד מזמור] intimates that the Shechinah rested upon him and then he uttered [that] song; ‘a psalm of David’ [מזמור לדוד] intimates that he [first] uttered [that particular] psalm and then the Shechinah rested upon him.
During the third meal on Shabbos afternoon [known affectionately as "shaleshudes"] we sing מזמור לדוד - which implies that the shechina is not yet upon us and only through our song it will descend. On Sunday morning in the shir shel yom we say לדוד מזמור [tehillim 24] which implies that the Shechina is already upon us and with that experience we begin to sing.
What is fascinating is that we see from here that Sunday morning contains a certain kedusha beyond and above most of Shabbos. On Sunday we are "on a roll" from the kedusha we felt yesterday that remains with us so we say לדוד מזמור. On Shabbos itself we apparently do not have this level of kedusha all day and we have to sing at shaleshudes [מזמור לדוד] to bring it upon us.
[שמעתי מהרה"ג רי"ד המניק שליט"א]
Let's appreciate Sundays known as the "Shabbos shniya shel galuyos" - The second day of Shabbos for Diapora Jews.
:-)
