I'm thinking of Robert Nozick's definition. He states his definition thus:

1. P is true
2. S believes that P
3. If it were the case that (not-P), S would not believe that P
4. If it were the case that P, S would believe that P

There is a reason why the Gettier rabbit-hole is so dangerous. You can always cook up an improbable counterexample to any definition.

For example, here is a counterexample to Nozick's definition as you present it. Suppose that I have irrationally decided to believe everything written in a certain book B and to believe nothing not written in B. Unfortunately for me, the book's author, a Mr. X, is a congenital liar. He invented almost every claim in the book out of whole cloth, with no regard for the truth of the matter. There was only one exception. There is one matter on which Mr. X is constitutionally compelled to write and to write truthfully: the color of his mother's socks on the day of his birth. At one point in B, Mr. X writes that his mother was wearing blue socks when she gave birth to him. This claim was scrupulously researched and is true. However, there is nothing in the text of B to indicate that Mr. X treated this claim any differently from all the invented claims in the book.

In this story, I am S, and P is "Mr. X's mother was wearing blue socks when she gave birth to him." Then:

1. P is true. (Mr. X's mother really was wearing blue socks.)

2. S believes that P. (Mr. X claimed P in B, and I believe everything in B.)

3. If it were the case that (not-P), S would not believe that P. (Mr. X only claimed P in B because that was what his scrupulous research revealed. Had P not been true, Mr. X's research would not have led him to believe it. And, since he is incapable of lying about this matter, he would not have put P in B. Therefore, since I don't believe anything not in B, I would not have come to believe P.)

4. If it were the case that P, S would believe that P. (Mr. X was constitutionally compelled to write truthfully about what the color of his mother's socks were when he was born. In all possible worlds in which his mother wore blue socks, Mr. X's scrupulous research would have discovered it, and Mr. X would have reported it in B, where I would have read it, and so believed it.)

And yet, the intuitions on which Gettier problems play would say that I don't know P. I just believe P because it was in a certain book, but I have no rational reason to trust anything in that book.

ETA: And here's a counterexample from the other direction — that is, an example of knowledge that fails to meet Nozick's criteria.

Suppose that you sit before an upside-down cup, under which there is a ping-pong ball that has been painted some color. Your job is to learn the color of the ping-pong ball.

You employ the following strategy: You flip a coin. If the coin comes up heads, you lift up the cup and look at the ping-pong ball, noting its color. If the coin comes up tails, you just give up and go with the ignorance prior.

Suppose that, when you flip the coin, it comes up heads. Accordingly, you look at the ping-pong ball and see that it is red. Intuitively, we would say that you know that the ping-pong ball is red.

Nonetheless, we fail to meet Nozick's criterion 4. Had the coin come up tails, you would not have lifted the cup, so you would not have come to believe that the ball is red, even if this were still true.