Yes, the first test you did is the AQ (Autism Quotient) test, and the second one is the Aspie test. What I've actually read that 32+ on the AQ test is likely Aspie, but the exact cutoff does vary a bit depending on where you look.
For me, I started with just my GP, who talked to me for about 10 minutes and then told me that I was showing some signs of it, then gave me a phone number I could call to find a psychologist. Talking to a GP really would be hit and miss - I had a really good experience and consider my GP to have been a very valuable step in my path to diagnosis, but I've heard of other people who have had a much worse experience.
Once I talked to a psychologist, he was able to tell me in a single session that he was pretty sure I have ASD, but that still wasn't a formal diagnosis - a gut feel from a psychologist isn't a diagnosis; that took about 5 more sessions before I got an actual diagnosis.
A formal diagnosis as an adult is quite expensive unless you have insurance that covers it. In Canada, it's not covered by MSP and would have cost me about $2400, and the psychologist did caution me that the formal diagnosis probably wouldn't get me a thing other than formal confirmation of what he already knew just by talking to me. I did have insurance that covered it, so I did proceed with the diagnosis.
So, my path to diagnosis was:
1) Someone telling me that he thought that I might have it (was a huge shock to me at age 36!)
2) Googling "Do I have Autism" and finding myself at the AQ test on Autism Canada's website, and scoring 37.
3) Talking to my GP and getting the initial feel that it was a possibility.
4) Talking to the psychologist and being told he was pretty sure I have it.
5) Formal assessment with the same psychologist.
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Level 1 Autism Spectrum Disorder / Asperger's Syndrome.