Why does Jesus refer to Himself as something distinct from God? This question is addressed to people of trinitarian sects. If Jesus is part of a trinity, why are there so many examples of Him referring to Himself as something distinct from God? Some examples are Him on the cross saying "My God, my God, why have You forsaken Me?" and when He refers to God as greater than Himself when speaking to the twelve prior to being taken into custody. There are more that I've noticed but these two come to mind first. Furthermore, Jesus is repeatedly said to sit at the right hand of the Father. Doesn't the phrase "sit at the right hand" imply that the Son is not equal to the Father?

I'm aware of there being counter-examples such as Him saying that He and the Father are one and of course, chapter one of John ("the Word was God").

Admitting these counter-examples support trinitarianism, how do Trinitarians explain the way Jesus speaks of God as if He is something distinct from God? Am I the only one who gets the impression that He speaks in this way?

The way I see it right now is that Jesus is the Father's proxy. All authority was given to Him to execute the Father's will. He was created by the Father (I've heard some say that He was "begotten, not made", but He is referred to as Firstborn of Creation) as God's self-expression or image (Col. 1:15). In this sense, He is a functional equivalent to the Father, but in another sense, He is not essentially equivalent because He came from the Father. Is this the same way Trinitarians see it?
Jesus is not the same person as The Father.

It is a key part of Trinitarian doctrine that the Father and the Son (and the Spirit) are distinct persons. Therefore is is completely correct and proper that Jesus talks about The Father as someone other than himself.

Likewise it is completely correct to call The Father "God" (just like it is completely correct to call Jesus "God") and so he can do that, although it is rarer than you might think - he usually refers to "The Father" when he is giving his own words and teaching, although he often refers to "God" when he is quoting scripture or the teaching of others.

Incidentally in the passage you quote Jesus is quoting scripture, so he would refer to "God".