Questions of Jewish status, especially in the contemporary context are not simple. I have been asked about Jewish status and conversion over a period of many years and would be glad to shed some light on the subject for your personal situation.
Let me be quite straight forward. Decades ago in Jerusalem having studied for the year in an Israeli yeshivah, I went to the Rabbanut—the Rabbinate to apply for what can be called a marriage license. I was told that I would have to bring two witnesses that know me from America and written proof that I was Jewish.
I thought that this was preposterous since I was studying in a yeshivah and studying to be a rabbi. If I’m not Jewish, why would I be studying in a yeshivah in Israel and to be a rabbi? The answer I would think is, that no one in today’s world can be too careful and it is required of all who want to marry religiously and most especially by the tenets of orthodox Judaism.
Parenthetically, one of my ei-dim (witnesses), who happened to be my roommate and knew me very well from both America and Israel, got so flustered under the rabbinic interrogation that he could not think straight and how to respond to the rabbis, testifying that he knew that I am Jewish!
Now to deal more directly to your question. I believe that you have accurately and honestly presented your religious status and that you continue to grow in your love and devotion to Judaism. It is understandable that you are somewhat uncomfortable with your true status in the eyes of Torah Judaism. You know that by the standards of many—other than Torah observant Jews—you would be considered Jewish with no further steps taken. This, however, is not our position rabbinically, which requires additional steps and a conversion—giyur through a recognized rabbinic court—beit din.
Given your obvious love of Judaism and devotion, the beit din in your locale should make the process rather easy and not as lengthy as a novice first approaching the topic.
Just as with other circumcised males who are not yet Jewish, a competent mohel—ritual circumciser will have to examine your milah—circumcision to be sure that it meets with Jewish custom, most likely requiring a ‘ha-a-ta-fat dam b’rit’—a drop of blood of circumcision and in the presence of three, forming a beit din. This requirement will be left to the discretion of the actual beit din.
My advice is to go to a rabbinic court with a high degree of acceptance so that you do not find yourself continually in a state of limbo, religiously, especially if you move to another area and will have to establish your status anew.
You will be required to immerse yourself in a ritualarium—a mikveh in the presence of the beit din and to place yourself under the ‘Yoke of the Kingdom of Heaven’—Ole Malekhut Shamayim, renouncing any allegiances to another god or religion.
Once you have properly been accepted, a ‘certificate of conversion’—te-u-dat giyur will be issued and signed.
Wishing you every success in your endeavors, then you will never be hesitant or uncomfortable to proudly proclaim your Jewishness.