Answer
Jesus meant He is greater than David.
Jesus said He is “greater than the Temple” (Mat 12:6); “greater than Jonah” (verse 41); and “greater than Solomon” (verse 42). He is greater than even patriarch Abraham because He was the I AM before Abraham came to be (John 8:58).
Jesus also meant that though the Christ was to be born in the lineage of David, the latter was not (“as was supposed” – Luke 3:23) the biological father of the Messiah. This was contrary to the Jewish belief. So OP’s attempt with Adonai/Adoni is irrelevant here.
Explanation
No father will ever call his son “my Lord”. Yet David called the Messiah, “my Lord”. Jesus says this shows the real Christ is much greater than King David.
Background:-
The incident where Jesus asks this question is found in all the synoptic gospels. So we need to take all of them into perspective to get the full impact of what Jesus meant.
In all the three gospels Jesus asks this question immediately after His discussion with the Pharisees regarding the greatest commandments in the Law.
Mark gives additional information. Jesus is questioned about the greatest commandment and He recites the famous Shema:
"Hear, Israel. The Lord our God is one Lord, and you shall love the
Lord your God with……” etc. (Mar 12:29-30).
“And the scribe said to Him, You say well, Teacher. You have spoken
according to truth, "that God is one," and "there is no other besides
Him;" (verse 32).
And Jesus also commends the scribe.
It is in this context that Jesus asked the question mentioned in the OP.
His question has several dimensions:
First of all, if, as the Jews believed, the Messiah would be a son of David, why would David call his own son “my Lord”? It doesn’t make sense. So it shows that the Messiah is more than the son of David.
It is well known to the Jews that King David called Yahweh Elohim as “my Lord and my God” (Psalms 35:23). David called Yahweh “my Lord” (Psalms 16:2; 141:8; 35:22).
So, Jesus’ question is aimed at knowing (in Psalms 110:1) who this “my Lord” of David is if not Yahweh Lord/Adonai!
- If, as seen above, David’s “my Lord” is Yahweh Adonai, then what about the Shema?!
Jesus knew the Shema:
“Hear O Israel, Yahweh our God is one Yahweh” (Deut 6:4).
Jesus also knew:
“And Yahweh God said, Behold! The man has become as one of Us”
(Gen 3:22).
The one Yahweh Elohim says “one of Us”!
The truth is, God is a composite one God/Elohim!
(No wonder, Moses used the uniplural word Elohim for true God throughout the Torah).
Conclusion
As a prophet of God, David knew the Messiah will be an Incarnation of Yahweh his Adonai. That is why he called Yahweh and the future Messiah as “my Lord/Adonai”.
Jesus implied that He is the Messiah, the “my Lord” of David and hence the Yahweh Incarnated. Jesus is God.
Post Script
Since the OP mentioned about the identity of the Adoni/Adonai, I have the following contentions to make:
I state that it is “Adonai” that was originally written in Psalms 110:1 and the supporting points are:
Point 1:-
Both “Adonai” and “adoni” looked the same in ancient Hebrew until the entrance of Masoretic Jews as the scribes in the Middle Ages.
Until then Hebrew was written only in consonants without any vowels. Vowels were used only orally when the OT was read.
Both “Adonai” and “adoni” look the same when written in Hebrew consonants. For example, if we remove the vowels from both “Adonai” and “adoni” in English itself, they will look “DN” and “DN”.
So, we have two historical considerations to make:
When was the vowels added in written form permanently in the OT?
In the Middle Ages when the Christians were stating that Jesus of Nazareth is the Messiah.
Who added the vowels permanently in the OT?
The Masoretic Jews who were familiar with the Christians who stated that Jesus of Nazareth is the Messiah.
So, chances are there they added the vowels of “adoni” instead of “Adonai”.
(I am in no way downplaying the exemplary service provided by the Masoretes).
Point 2:-
Who was David’s Lord?
David was a powerful king. He had no one above him except Yahweh. So let us see who was his Lord?
“Oh my soul you have said to Jehovah, You are my Lord” (Psalms
16:2).
“But my eyes are on You, O Jehovah, my Lord; in You I take refuge;
do not make my soul naked” (Psalms 141:8).
“You have seen, O Jehovah; do not keep silence, O Lord, do not
be far from me. Stir Yourself and awaken to my judgment, to my cause,
my God and my Lord” (Psalms 35:22-23).
[Does anyone remember Apostle Thomas declaring his faith in Jesus as “my Lord and my God” in John 20:28?]
It is very clear that Yahweh alone was David’s “my Lord”. There was no other real “my Lord” to David.
So, when David speaks “my Lord” in Psalms 110:1, it definitely is “my Adonai” not adoni.
Point 3:-
The right hand of a person in the Scripture is a very important place to be at.
This is all the more important when somebody is asked to “sit” at the right hand. Let us look at an example from the Scripture:
“And Bathsheba came in to King Solomon to speak to him for Adonijah.
And the king rose up to meet her, and bowed to her, and sat on his
throne, and placed a throne for the king's mother. And she sat at
his right hand” 1 King 2:19).
Bathsheba is no ordinary human to Solomon and Israel. She was the mother of the King. No king will allow any ordinary person to sit on a throne at his right hand unless that person is unique and special.
Then how could we ever think that the very God Almighty, Yahweh, will allow an angel (even an arch-angel) or any person to “sit” at His right hand?
No way!
Why?
Because, Yahweh’s “right hand is high” (Psalms 89:13) and “exalted” (Psalms 118:16) and no one can “sit” there, unless the Person is special or unique (Monogenes)!
If anyone has still doubt, see Whom King David has “set” at his right hand:
“I have set Jehovah always before me; for He is at my right hand:
I shall not be shaken” (Psalms 16:8).
So, the Scripture is very clear. A “lord” cannot sit at the right hand of Yahweh. So it has to be “Adonai”.
Point 4:-
The Greek translation of the OT is known as Septuagint which was translated around 250 years before Christ.
It is significant that the Greek translators used the same Greek word for both Yahweh and Adonai/adoni which is “KURIOS”!
“Kurios said to my Kurios”.
Unless they are crazy, no sensible persons would use the same word for the Almighty Yahweh and an ordinary person in the same sentence. They would have differentiated them.
The fact that they did not differentiate the terms confirms the truth that the second Hebrew word was “Adonai” in fact.
Point 5:-
“The Lord (Adonai) at your right hand” Psalms (110:5).
Who is this?
Verse 1 clearly “defines” and “introduces” the Messiah as the Lord/lord at the right hand of Yahweh. (This is the right hermeneutics, right?)
This cannot be anyone else because IF the Messiah is at the right hand of Yahweh, then Yahweh canNOT be at the right hand of the Messiah. Yahweh will be at the left hand of the Messiah!
So this Messiah who is the “my Lord/lord” of David IS the Adonai or the Lord at the right hand of Yahweh.
A likely objection:-
Some may argue that verse 5 is talking about Yahweh at “your” (that is, David’s) right hand.
But then there is a problem. Verse 7 talks about this same Adonai “shall drink out of the torrent on the way”!
This is talking about a triumphant Messiah who conquers kings and judges them.
If I am not wrong, Yahweh is never depicted as drinking from any torrent. (Objections are welcome).
So the “Adonai” in verse 5 and the supposed “adoni” in verse 1 are the same Person and the same Person is at the same right hand.
And this strongly points to “Adonai” to be the correct word in verse 1.
Adoni is used for human beings and Adonai is used for Yahweh. But in Psalms 110:1, strong chances are there, “Adonai” was used originally.
(Unless someone can disprove my above 5 points, I will stand vindicated).