I won't be so bold as to say who's right and who's wrong, but I will share my belief. I believe simply that God exists in three persons Father, Son and Spirit. I believe we can see this clearly in the scriptures with such passages as:
Genesis 1:26
26And God said, Let us make man in our image, after our likeness
Colossians 2
2That their hearts might be comforted, being knit together in love, and unto all riches of the full assurance of understanding, to the acknowledgement of the mystery of God, and of the Father, and of Christ;
3In whom are hid all the treasures of wisdom and knowledge.
In the above verse, we're ask to acknowledge God, Father, and Christ (God Holy Spirit, God Father, God Son). Jesus is God our creator, but when he walked the earth as a man, He subdued his divinity to walk as mere man. A mortal. He was our example. He prayed to God the Father as an example.
Mark 14:35-37
King James Version (KJV)
35And he went forward a little, and fell on the ground, and prayed ...
36And he said, Abba, Father, all things are possible unto thee; take away this cup from me: nevertheless not what I will, but what thou wilt.
Clearly we see two persons here. God the Son, praying to God the Father. Later Jesus told the disciples that He need to go otherwise the Comforter cannot not come.
John 16:7
7Nevertheless I tell you the truth; It is expedient for you that I go away: for if I go not away, the Comforter will not come unto you; but if I depart, I will send him unto you.
John 14:26
King James Version (KJV)
26But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.
In the above scripture we can clearly see the three persons of God. The Son speaking of what the Father will do, which is send the Holy Spirit.
The bible even tells us that Jesus is gone to Heaven and sits at the right hand of the Father. Two separate positions.
Mark 16:18-20
19So then after the Lord had spoken unto them, he was received up into heaven, and sat on the right hand of God.
One God in Three Persons. Holy Trinity.