Since I disagree somewhat with the other answer, I'll provide my own. You are pretty much correct in your view of them.
en/por la mañana
This is the probably the most common one you'll hear. Despite there being a small semantical difference in meaning (por meaning more like “at some point during”), regional usage seems to dictate usage far more than anything else.
de la mañana
Use this when referring to specific times like a las seis de la mañana.
You can also use it if you actually mean of/from the morning in English: “Es una criatura de la mañana / de la noche”. In many of those cases, you can replace with matinal/nocturna or similar for other periods of the day.
You can also omit the article (de mañana) to change the meaning to something close to "during the morning" or "come morning".
a la mañana
a la mañana will be a good bit less commonly used, though it has certainly been used in expressions like A la mañana al monte, a la tarde a la fonte for centuries. In this case, it might be best to think of a la mañana as a substitute for al amanecer (and a la noche as anochecer, etc). This usage survives today and seems to be picking up force, though what from, I'm not sure. It is not exclusive to any country, in fact, uses of it can be found on both sides of the Atlantic and both sides of the equator and sometimes can also have the meaning of by/come morning.
a la mañana, of course, can also be used in a more traditional sense if it ends up an indirect object such as in the title or if talking about things as if they were physical entities in the towards sense of a, for example:
Excerpt from “Faut-il s'abétir?” (Vicente Gaos, 1919-1980)
—¿Hacia dónde vamos?
—Vamos a la noche...
—¿De dónde venimos?
—También de la noche...