I think what is being conflated are the collection of the accounts from the different Gospels. In the original one (Mark), all it says is that the members that make up the Sanhedrin bound Jesus and presented him to Pilate. So, when Pilate asks him if he's 'King of the Jews' he's being asked a non-religious question. He's essentially being asked if he believes himself to be rightful ruler instead. It only makes a passing mention of questions by the priests (which would not happen in Pilate's court). I can see there be some reconciling between the Synoptic Gospels, but John's doesn't seem to fit very well.
John 18:32 says that Pilate tells the priests they should judge Jesus according to their laws. So, he's brought in front of Pilate, he defers to them (would never happen). They answer that they have no power to put a man to death (wrong; they could if he were convicted of heresy). Regardless, he only starts his own judgment after deferring to them (again; would never happen, especially by Pilate in particular). 18:35-39 makes no sense whatsoever. Verse 37 in particular has him admit that he's a king, though "not of this world". So, whoever wrote these verses did not know enough about Jewish religious law because they didn't know that the Sanhedrin indeed could sentence a man to die by stoning. There's no mention by Pilate that he went against Roman law (Pilate even says that "I find no fault at all"). So, why wasn't he released? If the priests couldn't sentence him and only Pilate could and he didn't find him guilty, why was there even a crucifixion? Instead, he's still being held against his will. Then (verse 39), the "tradition" (that never existed), about releasing a prisoner. This was never a tradition either in Jerusalem nor in Rome. So, if all the Gospels agree with one another, how is this reconciled?