Was Hermione mistaken about her and Harry's DADA exam history?

Question

This discussion regarding Defence Against the Dark Arts results takes place in chapter 15 of Order of the Phoenix, emphasis mine.

“Me?” said Harry, now grinning more broadly than ever. “No I’m not, you’ve beaten me in every test —” “Actually, I haven’t,” said Hermione coolly. “You beat me in our third year — the only year we both sat the test and had a teacher who actually knew the subject. But I’m not talking about test results, Harry. Look what you’ve done!”

This is absolutely a nitpick, but it got me thinking: isn't Hermonie's last condition - "and had a teacher who actually knew the subject" - redundant? To my memory, third year is the only year where both Harry and Hermione sat their DADA exam. As I recall:

• In first year, Dumbledore cancelled the exams as a school treat. And as I recall it, something about Quirrell being a good teacher is what started this conversation.
• In second year, Hermione missed them due to being petrified.
• Third year isn't a problem.
• In four year, Harry was exempt due to the Triwizard tournament.
• Any exams after fourth year haven't happened yet.

Am I mistaken? Or was third year the only year where they both sat their DADA exam?

Answer 1

It seems you are mixing up the first and second year. It was the second year that the exams were cancelled. From Chapter Eighteen of Chamber of Secrets:

or Professor McGonagall standing up to tell them all that the exams had been canceled as a school treat (“Oh, no!” said Hermione),

In the first year they took all their exams. From Chapter Sixteen of Philosopher's Stone:

Their very last exam was History of Magic. One hour of answering questions about batty old wizards who’d invented self-stirring cauldrons and they’d be free, free for a whole wonderful week until their exam results came out. When the ghost of Professor Binns told them to put down their quills and roll up their parchment, Harry couldn’t help cheering with the rest.

Thus, in their first year they had both taken the Defense Against the Dark Arts exam. The unique situation of their third year was that in addition to both of them taking the exam they also had a teacher who knew the subject.