Here in Goodfellas, Tommy was killed in his making ceremony, and to me, something was unclear. Who were the true authority that was supposed to make him ? What the thing Paul and Vinnie had to do with it ? The old man, who stood behind Tommy when he was shot, was he Vinnie ? Did Paul give the go ahead and if not why he took no move as a vendetta ?

Tommy wasn't being made. It was a trap. They used the making ceremony as a cover. I don't know about the other part of your question though.

I also do not know much or less than you, I see.

Paulie was the boss--he could order others to do his killing for him. The old guy, Vinnie, was there because he looked harmless, and Tommy wouldn't suspect him of foul play.
Also, the man who played Vinnie was Martin Scorsese's father.

Nice interpretation, let me ask again though -
So, that old man during the hit was Vinnie ?
And Paulie ordered the hit ?

Yes to both.
Batts was a member of the Gambino family. Paulie was a member of the Lucchese family. But since one of Paulie's people (Tommy) killed Batts, the Gambinos had the right to demand that Paulie order the murder of his own person to satisfy the Gambinos' need for vengeance.
You might wonder: Why wouldn't Paulie simply let the Gambinos take care of whacking Tommy? The answer is: Tommy was Paulie's responsibility. And, if Paulie let another family whack one of his own people, it would set a bad precedent. Paulie would lose face, then lose power.

The interpretation is so beautiful, how did you get that ? From the novel ?

Thanks. What I wrote is simply Mob protocol: someone in a boss's crew is his responsibility. Allowing someone from outside the family to settle that responsibility is a mark of weakness.

In the film, both Jimmy and Henry were complicit in Billy Batts' murder. But Paulie didn't punish them. We might think that the reason was because Jimmy wasn't Italian and Henry was only half-Italian, so they were ineligible to be "made," and therefore couldn't be expected to live by the "rules," as Tommy was expected to. By far the more likely reason is that Tommy was a "cowboy," as Paulie explained to Henry earlier. He was wild, dangerous, not to be trusted. A guy who would kill a made man from another family in a personal dispute was a liability to Paulie. And, even more important: Jimmy and Henry were "good earners." Paulie needed the money they brought in.

It is even nicer. Thanks, I got it clear now.

Batts was also a friend of John Gotti.