learning to be a better liar
Of-course. Everybody wants to be liked and may be he is trying to improve, be positive.
And you know this how? Did he leave the web-page open "by mistake"? Or did you check out his browser history in exactly the same way you discovered his lying... since he didn't learn from his mistake or use it to manipulate you or anything.
While I'm skeptical without some example from him to validate redemption, I do believe people are good and can recover from a moral lapse with enough effort.
No, your situation is unlike what I am familiar with.
People can change but it really depends on the person and how much you are willing to give them another chance. Admitting you have a problem is the first step in correcting it. If he wants to fix who he is, he would have to either admit he has been lying to you and write off your relationship and start over with someone else and never lie to them.
If your heart no longer trusts him, it doesn't matter if he is telling the truth or lying. You just don't trust him. I would tell him the truth and move on. The is no way to build a relationship if there is no foundation of honesty and trust.