There is already quite some discussion about the (final) name of the site, but to me this seems somewhat backwards. I think we should start with sharpening the target audience, looking at the question themselves of course.
First of all, it is a good point that every Stack Exchange site includes a short blurb describing the target audience. Here are a few:
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.
MathOverflow is a question and answer site for professional mathematicians
Mathematica Stack exchange is a question and answer site for users of Wolfram Mathematica.
Academia is a question and answer site for academics of all levels.
It seems to me that a good one for this site would be something like:
[Name to be determined] is a question and answer site for mathematics teachers, math education researchers, and anyone interested in the process of teaching or learning mathematics.
It's true that it feels premature to talk about this sentence before deciding on a title. On the other hand, I discussing this sentence might be helpful for choosing a title, because it clarifies the different ways that we all think about this potential site.
It seems that one main point of contention is whether we want students on this site. My answer to this question is yes, but only if they want to talk about teaching, curriculum, or pedagogy. I can certainly imagine student questions that would pertain to these topics, such as:
Why do we learn straightedge and compass constructions in geometry class?
Why is it important for me to learn to do math without a calculator?
Why are real analysis and abstract algebra required for a math major?
Is there a good way to remember all of these logarithm rules?
Is it better to take AP Calculus or AP Statistics?
How can I study for calculus tests more effectively?
(Note: If you like one of these questions, feel free to post it as an example question on the main site.)
At the same time, I think it's much more common for students to have questions about math itself, as opposed to the process of teaching or learning mathematics. Any question that's primarily about mathematics should be migrated to the Mathematics Stack Exchange, and it's an important practical goal to minimize the frequency of such questions. As I have stated elsewhere I think this means that the title of the site should be very clear about the type of questions we're looking for.
In summary, I don't think that we should exclude students (or any interested audience), but I do think it's very important to exclude certain types of questions.
It seems to me the example questions are mainly questions asked from a teachers perspective, and I think we should really focus on this. Otherwise, that is having many questions about how do I go about learning/understanding something and so on, it will be much harder to keep up a meaningful distinction to the actual mathematics sites.
In my opinion the target audience should be
Teachers of mathematics (at any level) and researchers in mathematics education
This does not mean I want to exclude anybody in a strict sense. And, the goal in changing the target audience is really not so much to be exclusive, only experience from other sites tells me that one should be rather more strict in the description. Interested students and however else interested, are certainly welcome, but regarding content the thus declared target audience will provide an additional guideline regarding the type of question welcome on the site.
I can understand your logic, but at the same time, if you cannot describe what you want in a very concise "elevator pitch" — with an easily-understood title and simple description — you'll likely have a lot of trouble gathering support.
Coming up with a title and description of what you envision is step one. If you're just throwing everything into a big pile (like the proverbial "kitchen sink"), the name you try to come up with later to capture it all will be horribly convoluted with a gerrymandered scope that will never gather support. That's not the way to rally folks around an idea.
You almost have to come up with the title first to let people rally around… something. You can get it wrong and fix it later, but if you're starting with the premise, "we don't know what we want, but we'll figure that out later," that's not the way to pitch a "product" … and it makes it even harder to attract an audience.