Get my logo symbol from my logo to use in business cards -- 2

My logo is currently as a png file and I am looking to get the logo symbol from it in vector form to use on business cards. Specifically, I would like to have the logo symbol in a smaller size but with the same colors as the original logo. Additionally, I have a concept of how I would like the logo symbol to look on the business cards, so I will provide you with guidelines to follow.