Hindi ko malaman kung paano ang magiging approach ko sa sagot.
I do not believe in the Trinity. The word "Trinity" does not appear in the bible. The concept of a Trinity is not biblical.
Your question is premised on a belief in the Trinity doctrine. So, if I disagree with the premise, I can't answer without first changing the premise.
Pero puwede rin siguro, kung limited lang sa ganitong issue ang sasagutin ko:
What should have been the reason of such designation when only one God can actually do anything he wants as it pleases himself?
If God can do anything He wants, then He does not need a reason for doing what He does.
God has revealed some things to man, but He has not revealed everything to man. All we can do is understand the things revealed. We can't understand things that were not reveled.
For example, God revealed that the sins of all mankind were forgiven through the death of Jesus. Why did Jesus have to die for the forgiveness of sins? Because death is required as sacrifice for the atonement of sins. As a final sacrifice for all, the sacrifice should be a man without sin. Since Jesus is the only one without sin, then He is the only one qualified to become a sacrifice for the forgiveness of sins.
That's the extent of what was revealed.
We might ask --- why is a sacrifice even necessary? If God is all-powerful, then surely He can forgive all sins if He wants to, without need for any human sacrifice. Why the need to go through all this trouble?
There is no answer to that question, because it is beyond the scope of what was revealed to us.
In the same manner, if we go to the question of why God wanted the designations of Father, Son and Holy Spirit, when it would have been much simpler to have one single God in one person, the same principle applies.
God revealed the existence of the Father, the Son and the Holy Spirit, and that is the limit of what was revealed. Why didn't God just make it simpler? There is no answer to that question.
We should not apply our own logic, because we will still fail to understand. That's why God explains:
“As the heavens are higher than the earth, so are my ways higher than your ways and my thoughts than your thoughts. (Is. 55:9)