In general, which one is conveys a stronger conviction/opinion? I did some quick google search but didn't find any interesting results.
I'm positive that
I strongly believe that
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"I'm positive" is typically used for factual questions and means there is no doubt whatsoever. "Are you sure that you have enough money to pay for a sandwich?" "Yes, I'm positive, I have two thousand dollars in my pocket!" or "No, I'm pretty sure, but I'm not positive, let me count."
By contrast, "I believe" is more typically used for opinions. "I believe" expresses an opinion and "I strongly believe" simply makes the opinion more strong (just like it says).
You can also use "believe" for a fact. If you use "believe" for a fact, rather than an opinion, then it's inherently weaker than being positive. You can say for instance, "I believe that I am older than you, but I'm not very confident in that belief." Again putting "strongly" here increases certainty. However, "positive" means that you are 100% certain.
The word "positive" can also be used to mean "optimistic," but typically means that one is optimistic in disposition, not about a particular opinion or fact.
When you are positive about something, you are optimistic. You are pretty hopeful or expect that to happen.
When you strongly believe it, it's more of your stance/confidence with affirmation.
If things don't happen the way you thought.
In case where you were being positive will simply accept it as it was just your hope and expectation. There are still chances that next time, you may still give it a shot and again be positive.
In case where you strongly believed, you'll lose confidence or faith in that belief. There are less chances that you again believe the same story and go for it in the future.