Nagalinganesh is Hindu Trinity

My colleague wanted to have mini Ganesh idol similar to the one I posted earlier.

Out of over confidence I said I am going to make a better one for him. And ended up with an idol which is a combination of Ganesh+Shiv lingam+Serpent.

It looks to me like a Hindu Trinity now. How?
   Ganesh=Son
   Shiv lingam=Father
   Serpent=Holy Ghost

Better I make a child Ganesh that he wanted. I am in dilemma to gift this sort of unusual forms to anyone.