First some terminology. To convert a prospect to a member, all you do is add a Program to them. Programs are sometimes called agreements or contracts in some businesses. To add a program, go to the prospect's detail page, and click the button "Convert to Member." It is as easy as following the steps presented to you.
To Add a Program to an existing Member, go to the Member's Details Page, and in the Quick Action drop down select "Add a New Program" and click the green "Go" button.
We calculate the New Member Stat on the dashboard by counting how many Programs (contracts, agreements) have been entered during this month.
We use the "Agreement Signed Date" you entered while adding the Program. It is not based on the start date of the agreement.
Another place that can prevent the tracking of a contact as a New Member is the "track as renewal" checkbox while adding a program.