Can I call a question like
Do you want a red or blue eraser?
an 'alternative question' or should I call it a 'question with two alternatives'?
You can call them binary questions.
binary adjective Relating to, composed of, or involving two things. ‘To avoid the binary thinking that collapses complexity, it is necessary to assess both similarities and differences while watchful for the excesses of either prejudice.’ - ODO
Here are some usage examples:
A binary question is one with two possible answers, such as yes or no, or equivalently, true or false, or even one or zero. - Quantum Structural Studies
Use Binary and Non-Binary Questions When Selling - Michael Angelo Caruso
What is a binary question? It is a question that only requires only two possible answers. (Example:Yes or No) - answers.com
Regarding whether such questions may be called alternative questions or questions with two alternatives: the latter is fine, but the natural reading of the term alternative question is another question that one might answer instead of the 'primary' question. For example, in a guessing game that has a question that is difficult to answer, you might have an alternative question that is easier to answer but gives you fewer points.
If I had to pick one from the two you provided, I'd go with 'alternate question'. The second one, is however very tricky and can be ambiguous.
If you tell me 'a question with two alternatives', I can either interpret it as:
a question with two solutions or options.
or
a question, with two alternative questions.
You could even try multiple-choice question, but then again, that's a question with more than one solution, and can have two, three or any number of choices.
Or you could simply call it 'a question with two choices'.