The woman says "my beloved" and "I love" when she talks about a shepherd:

> Tell me, O you whom I love, Where you feed your flock, Where you make
> it rest at noon. For why should I be as one who veils herself By the
> flocks of your companions? (1/7)
> 
> My beloved is mine, and I am his. He feeds his flock among the lilies
> (2/16)
> 
> My beloved has gone to his garden, To the beds of spices, To feed his
> flock in the gardens, And to gather lilies (6/2)
> 
> I am my beloveds, And my beloved is mine. He feeds his flock among the
> lilies (6/3)

But when she mentions Solomon, she doesn't say "beloved" or "I love"

> Draw me away! We will run after you. The king has brought me into his
> chambers. We will be glad and rejoice in you. We will remember yyour
> love more than wine. Rightly do they love you (1/4)
> 
>I am dark, but lovely, O daughters of Jerusalem, Like the tents of Kedar, Like the curtains of Solomon (1/5)
>
> While the king is at his table, My spikenard sends forth its fragrance
> (1/12)
> 
> Behold, it is Solomońs couch, With sixty valiant men around it, Of
> the valiant of Israel. They all hold swords, Being expert in war.
> Every man has his sword on his thigh Because of fear in the night. Of
> the wood of Lebanon Solomon the King Made himself a palanquin: He made
> its pillars of silver, Its support of gold, Its seat of purple, Its
> interior paved with love By the daughters of Jerusalem. Go forth, O
> daughters of Zion, And see King Solomon with the crown With which his
> mother crowned him On the day of his wedding, The day of the gladness
> of his heart (3/7-11)
> 
> Solomon had a vineyard at Baal Hamon; He leased the vineyard to
> keepers; Everyone was to bring for its fruit A thousand silver coins
> (8/11)
> 
> My own vineyard is before me. You, O Solomon, may have a thousand, And
> those who tend its fruit two hundred (8/12)

Also, if the beloved was the king Solomon, daughters of Jerusalem wouldn't ask "what is your beloved more than another beloved?". The question implies the beloved is a simple man rather than a king

Also, it seems to me that here she is differentiating between the beloved (second person) and the king (third person):

>Draw me away! We will run after you. The king has brought me into his chambers

It seems she is asking him to take her away from the king's harem

In Chapter 3, we see that she brings her beloved to her home and then Solomon's couch arrives which also suggests that they are different people. The beloved is at her home while Solomon is on his couch at the time:

> By night on my bed I sought the one I love; I sought him, but I did
> not find him. «I will rise now,» I said, «And go about the city; In
> the streets and in the squares I will seek the one I love.» I sought
> him, but I did not find him. The watchmen who go about the city found
> me; I said, «Have you seen the one I love?» Scarcely had I passed by
> them, When I found the one I love. I held him and would not let him
> go, Until I had brought him to the house of my mother, And into the
> chamber of her who conceived me. I charge you, O daughters of
> Jerusalem, By the gazelles or by the does of the field, Do not stir up
> nor awaken love Until it pleases. Who is this coming out of the
> wilderness Like pillars of smoke, Perfumed with myrrh and
> frankincense, With all the merchant́s fragrant powders? Behold, it is
> Solomońs couch, With sixty valiant men around it, Of the valiant of
> Israel. They all hold swords, Being expert in war. Every man has his
> sword on his thigh Because of fear in the night. Of the wood of
> Lebanon Solomon the King Made himself a palanquin: He made its pillars
> of silver, Its support of gold, Its seat of purple, Its interior paved
> with love By the daughters of Jerusalem. Go forth, O daughters of
> Zion, And see King Solomon with the crown With which his mother
> crowned him On the day of his wedding, The day of the gladness of his
> heart (3:1-11)