In the beginning of Harry Potter and the Chamber of Secrets Harry scares Dudley into thinking that he is allowed to do magic out of school. However this should have never scared Dudley as Petunia should have known from Lily's time as a student that students aren't allowed to perform magic out of school. So why is it that Petunia didn't know or perhaps didn't tell Dudley?

Because, according to Petunia herself, Lily didn't obey that rule.

"Knew!" shrieked Aunt Petunia suddenly. "Knew! Of course we knew! How could you not be, my dratted sister being what she was? Oh, she got a letter just like that and disappeared off to that - that school - and came home every holiday with her pockets full of frog-spawn, turning teacups into rats..."
_{Harry Potter and the Philosopher's Stone, pg.44}

Bellatrix's answer cites an FAQ on Rowling's website, in which she confirms that while Petunia is "exaggerating a little", Lily really did use magic outside of Hogwarts:

Aunt Petunia is exaggerating a little; you have to allow for her state of mind when she started shrieking these things. However, just like her son, Lily was not averse to testing the limits of the Statute of Secrecy, so you can safely assume she will have had a few warning letters – nothing too serious, though.

Petunia must not have been aware of the warning letters or their contents, and given she claims her and Lily's parents were "proud" of having a witch in the family, her parents may not have punished Lily even if they were aware of the letters. As a result, Petunia never learned about the rule and assumed wizards were allowed to use magic as they pleased.

It’s still possible for Harry to use magic.

Though it’s against the rules for an underage student to use magic outside of Hogwarts, they absolutely can and do despite that. Petunia’s sister Lily had used magic at home, so she’d have little reason to believe that Harry wouldn’t either. Petunia told Harry that his mother turned teacups into rats every holiday, and there was some truth to what she was saying. In the F.A.Q. section on her website, JKR answered why Lily wasn’t expelled from Hogwarts despite Petunia saying she was turning teacups into rats at home, and in answering that, she also confirmed that Lily did in fact use magic outside of Hogwarts.

In "Philosopher's Stone" Aunt Petunia says that Lily came back from Hogwarts with frog spawn in her pockets and turned teacups into rats. If this is true, why wasn't Lily expelled?

Aunt Petunia is exaggerating a little; you have to allow for her state of mind when she started shrieking these things. However, just like her son, Lily was not averse to testing the limits of the Statute of Secrecy, so you can safely assume she will have had a few warning letters – nothing too serious, though.

If Petunia was using what she learned living with her sister to inform her judgments, she’d have no reason to not consider it a possibility for Harry to use magic outside of Hogwarts. Harry actually does come close to using it on Uncle Vernon.

“But a reckless rage had come over Harry. He kicked his trunk open, pulled out his wand and pointed it at Uncle Vernon.

‘She deserved it,’ Harry said, breathing very fast. ‘She deserved what she got. You keep away from me.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 2 (Aunt Marge’s Big Mistake)}}

Therefore, Petunia had good reasons to believe that it was a possibility that Harry could use magic on them even though it was against the rules for him to use it outside of Hogwarts.

I think the key distinction here is the difference between knowing Harry can't and knowing that he shouldn't.
The latter is the reality since Harry is physically capable of performing magic out of school (as we see in 'Order of the Phoenix'), it just carries with it consequences.
On top of that Dudley is a coward and has been on the receiving end of magic from both Harry and Hagrid. So his reaction is consistent with his character and what he knows, or at least believes.