Hi,
I agree with the other contributors. Let me just add my two cents.
First, you didn't tell us how you are feeling. What are the reasons that made you begin therapy? What are your symptoms? You may or may not have depression. The only person who can give you a diagnosis is a doctor (as you state, your counsellor may have an opinion, but not a diagnosis). It would be better to see a specialized doctor (i.e. a psychiatrist), but you may start by talking to your GP, who will then refer you to a psychiatrist if need be.
Second, if you feel ill, you go to the doctor: that's how easy it is. It doesn't matter whether your symptoms are in your stomach, in your eyes, or in your brain. A doctor's duty is to help you get better independently of what illness you have, so you should not be ashamed to go and visit a doctor about your symptoms, the same way you wouldn't be ashamed to go to the doctor for a stomach flu or an aching back.
Third, when you see a doctor for any kind of sickness, they will typically give you medications. And you will typically take what you're prescribed. Some meds work right away, some take a little longer. IF (big if) you are diagnosed with depression, you will most likely be prescribed medications that take a while to work and you will have to take them for quite some time. This does not make them more toxic or more dangerous than any other medication you have taken in your life. Nobody "likes the idea of being on medication", but I guess all of us would rather take medications than be sick.
Fourth, I understand that talking about feelings and thought and other personal stuff may not be the easiest thing for some. When you go to a doctor, you should not think of yourself as telling them about feelings, but about symptoms. You are not going to the doctor to tell them that your heart is broken or that you are unhappy with your family situation (or stuff like that). You are going to the doctor with a list of symptoms that you believe constitute a health problem (whatever symptoms you have). Yes, the doctor (a psychiatrist in particular) may ask you personal questions to "contextualize" your problem, but you will find that the kind of conversation is quite different (less personal, more clinical) that those you have with your counsellor.
Finally, while I understand your difficulty in sharing personal feelings with people, I think that talking to friends or family (depending on your situation) is of great help in a situation of psychological distress of any kind. I am sure you have someone who is ready to listen to you and I think you may be greatly surprised by the ability of our loved ones to listen and try to understand / relate to your situation, even if you think that your current feelings are "not normal" or bizarre.
Keep us updated.