There are those who will tell you that you are not Wiccan because you have not been properly initiated into a coven with a proven initiatory lineage leading back to one of the New Forest Covens, such as the coven which initiated Gerald Gardner.
There are some very good reasons why this is so. Traditional Wicca isn't about what you believe or whether you follow the Rede. The Rede isn't even part of British Traditional Wicca. What makes one actually Wiccan in their viewpoint is that one practices in a specific way as passed down from initiate to initiate, and that one worships a specific God and Goddess whose very names are oath-bound and no given to non-initiates. So in the viewpoint of those who hold to the original definition of Wicca, anyone who is not an initiate of British Traditional Wicca is not a Wiccan.
Personally I believe that the term "Wicca" has long escaped from that sort of limited meaning and has come to include those who learn from authors such as Scott Cunningham and may never undergo initiation at any time.
I don't believe that one has to do a self-dedication to consider yourself Wiccan; however, if you are serious about your religion then a dedication can be a powerful ritual commitment on your part.
Where I find I have more issue is when people call themselves Wiccan because they read the good parts of one book and dress all in black but don't actually practice the religion. They don't do regular honor to the Gods and Goddesses, they don't mark the Sabbats and Esbats, and they don't truly look at this as the spiritual journey that it is. They are what I call "Playgans", playing at something which should be important to who they are.
I your case, if you are serious about what you are doing and if it is filling an important spiritual place in your life, then I don't think you need to worry about whether you are a "real Wiccan" or not. It's quite enough that you know what is in your heart.