What is God's definition of forgiveness? I have wrestled with this topic for years and have come to all sorts of conclusions along the way. Please, I'm searching for what others perceive is God's definition of forgiveness. 

If we are wronged terribly by someone, is it "ok" if there is always a bit of us that is guarded when it comes to talking/thinking/interacting with that person? 

We may not feel anger or wish them any harm, but as a human being it sometimes feels almost impossible to forget the hurt that has been caused by someone else.
Jesus said to be "as wise as serpents, but as gentle as lambs.". Being a Christian does not mean checking our brains at the door when it comes to interacting with the world. Indeed, we are often called to be "discerning."

My interpretation of that word is this - forgiveness doesn't mean forgetfulness. 

We can still be guarded when dealing with others, and if they are looking like they are truly changing, give them more trust. When John Mark left Paul, for example, Paul initially said not to trust him. It wasn't until the end of his ministry that Paul changed his mind, mostly on the evidence of a track record. 

Forgiveness would imply giving people a chance to make amends - to be different - but it doesn't imply an open invitation for somebody to demand trust that is not warranted by their behavior.