Discrepancies between the gospels on the account of John

Luke mentions Peter visiting the empty tomb, but does not mention John. If the disciple Jesus loved was John, shouldn’t he have been mentioned too in Luke?

Luk 24:12 Then arose Peter, and ran unto the sepulchre; and stooping down, he beheld the linen clothes laid by themselves, and departed, wondering in himself at that which was come to pass.

Why wasn’t Lazarus mentioned? I think the same reason he was not mentioned at all in any of the synoptic gospels – he was a youth and considered to be inconsequential.

In Mark, the 11 apostles are described to have unbelief and hardness of heart until they actually saw Jesus.

Mark 16:14 Afterward he appeared unto the eleven as they sat at meat, and upbraided them with their unbelief and hardness of heart, because they believed not them which had seen him after he was risen.

However, in John 20:8, the other disciple believed. What did he believe? He must’ve believed that Jesus was resurrected.

Jhn 20:8 Then went in also that other disciple, which came first to the sepulchre, and he saw, and believed.

If the disciple was John, then it would be inconsistent with Mark 16:14.

Why would Lazarus immediately believe? Because he saw the graveclothes and it reminded him of his own resurrection.

Jhn 20:7 And the napkin, that was about his head, not lying with the linen clothes, but wrapped together in a place by itself.

Jhn 11:44 And he that was dead came forth, bound hand and foot with graveclothes: and his face was bound about with a napkin. Jesus saith unto them, Loose him, and let him go.