Do wizards get a diploma when they graduate from Hogwarts? Does it say "professional wizard" or something?
There's no record of any diplomas. Graduates are described as "qualified wizards".
The phrase qualified wizard is the only term I can find which distinguishes wizards and witches who've finished their education from Hogwarts students and underage children. It's used to describe those who have finished their education...
"I must warn you, Harry, that the charm might be too advanced for you. Many qualified wizards have difficulty with it.”
(Prisoner of Azkaban, Chapter 12, The Patronus).
Mr. Weasley was a fully qualified wizard who worked in the Misuse of Muggle Artifacts Office at the Ministry of Magic...
(Goblet of Fire, Chapter 2, The Scar).
“I am a fully qualified Ministry official, Malfoy, do you really think I cannot manage two wandless teenagers alone?” asked Umbridge sharply.
(Order of the Phoenix, Chapter 32, Out of the Fire).
...and those who have not...
Hagrid wasn’t a fully qualified wizard; he had been expelled from Hogwarts in his third year for a crime he had not committed.
(Prisoner of Azkaban, Chapter 5, The Dementor).
“And you think you’ll be able to do something for him?” asked Zabini scathingly. “Sixteen years old and not even fully qualified yet?”
“I’ve just said, haven’t I? Maybe he doesn’t care if I’m qualified. Maybe the job he wants me to do isn’t something that you need to be qualified for,” said Malfoy quietly.
(Half-Blood Prince, Chapter 7, The Slug Club).
“I do not think you will count, Harry: You are underage and unqualified. Voldemort would never have expected a sixteen-year-old to reach this place:"
(Half-Blood Prince, Chapter 26, The Cave).
“Nice job, I hope? Pleasant? Easy? Sort of thing you’d expect an unqualified wizard kid to be able to do without overstretching themselves?”
(Deathly Hallows, Chapter 28, The Missing Mirror).
Students can get additional awards, like the Award for Special Services to the School, although those obviously aren't given to most. There were awards like the Barnabus Finkley Prize for Exceptional Spell-Casting back in Dumbledore's day, but it's unclear whether or not these still exist.
Now approaching his eighteenth birthday, Dumbledore left Hogwarts in a blaze of glory - Head Boy, Prefect, Winner of the Barnabus Finkley Prize for Exceptional Spell-Casting, British Youth Representative to the Wizengamot, Gold Medal-Winner for Ground-Breaking Contribution to the International Alchemical Conference in Cairo.
(Deathly Hallows, Chapter 18, The Missing Mirror).
There's no evidence of any specific award or qualification that's gained by more contemporary students on leaving school (although, obviously, they gain O.W.L.s and N.E.W.T.s). The fact that you have completed your education is enough for society to view you as a qualified wizard.
JK Rowling had discussed graduation ceremony in one of the podcast but nothing was discussed about diploma.
John Noe: You know, what I'm curious about now. What I think is one of the neatest things about the Hogwarts tradition is the entrance ceremony, from the whole riding the boats to the castle to the Sorting ceremony. What kind of traditions is there for graduation and leaving Hogwarts?
J.K. Rowling: D'you know, John, I'm really glad you asked that, because I felt a huge sadness that I wouldn't write a graduation scene. You know, I really did. I knew-- I mean, I knew from early days that we would never see them graduate. I knew that he would-- well not he, they, all three of them, would not. We would not see them at school during what would've been their final year of education. But I really, during the final book, I kept thinking it would've been-- I felt sad that the book wasn't gonna end with that Feast scene, the graduation scene. But it couldn't, you know, it just couldn't. That's not the way it could've ended. It would've felt far too trite and-- you know, a lot of people felt the Epilogue was too sentimental, I think to have a graduation scene on top of what just happened would've been an absurd bit of anti-climax.
John Noe: Did you have ideas for what kind of traditions that they would do? Like ride the boats back out of Hogwarts, obviously, I think it's the cutest thing...
J.K. Rowling: Oh yeah, definitely. No, I think the boats would've been the most poetic and beautiful way to get-- for them to leave. And symbolic in that they-- Harry wouldn't have seen the thestrals again, you know what I mean? It would've been a return to innocence, really. And passage over water is so symbolic, you know, in the history of magic, so, yeah. That would've been great.