Given what I am, should I be a computer scientist or software engineer?

I appreciate your enthusiasm, but it seems to me that you are worrying to much. I seem to understand from your Quora profile that you are still in high school, which means you are at least 4 years away from being in a situation where your decision of being a computer engineer vs a computer scientist is going to matter for what you are going to do next (namely go to graduate school vs start working). You also seem to be overstating the difference between computer engineers and computer scientists. Rest assured that if you are a good enough engineer you will likely face entirely new theoretical problems, and that as a computer scientist you can choose to study a field that will require plenty of engineering work. Take for instance research in networking. While plenty of the work in that domain is done by computer scientists at universities, a significant portion of it (especially research regarding data centers) is done by companies that have an immediate need for it (Google, Facebook and Amazon have actual data centers, while university researchers mostly speculate about them), and many of the people working on these problems at companies are engineers and have an engineering approach to solving them. In any case, you'll have plenty of time to choose and most importantly change back later, for now I would advise you to study computer science at the best school that will accept you until you get a bachelor's, and then you can reconsider what you want to do next. I also wouldn't worry too much about not being able to make it as a computer engineer because you didn't know how to write an OS when you were in high school, I don't think that was the case for many of us either. And your English is pretty decent by the way, you know, plenty of papers written by foreigner contain weird English, and plenty of papers in general contain grammatical mistakes or typos. I'm not saying you shouldn't avoid them, but it isn't that big of a deal.

There are algorithmic software engineers and computer scientists who develop software systems.  And have no doubt: a weak command of written language is no obstacle to being published. Ultimately, you need to develop a specific skillset, and then find someone who's looking for what you do.  Don't worry too much about the label you attach to yourself just yet.

Just do what you feel like. Pick a major as late as possible, and by the time you absolutely must it'll be obvious. And you can be a software engineer who never uses math or one who uses math every day at extremely high levels, depending on what you're engineering. Relax. Stop thinking these descriptors mean anything. Do what you like at each point and it'll be fine.