“Alternately” or “alternatively”

Question

What is the difference between alternately and alternatively? I've seen both words being used, but which one is grammatically correct?

He could do X. Or alternately, he could do Y.
He could do X. Or alternatively, he could do Y.

Answer 1

Alternately means switching between two alternatives, alternatively means doing something different.

I love pizza and Mexican food, I eat them alternately.

Means Monday I ate pizza, Tuesday tacos, Wednesday pizza again, Thursday burrito, Friday went to the Hospital to get my arteries roto-rooted.

I love pizza but alternatively I eat Mexican.

Means both choices are good, but says nothing about when or the order in which I eat them.

Alternately is about switching. Alternatively is about enumerating options.

Answer 2

Fraser Orr's answer corresponds to how I learned English in the UK in the 50's and 60's.

However, the Oxford English Dictionary, under 'alternate', gives its first six meanings consistent with this, but then it says

II. Senses equating to alternative adj. Chiefly N. Amer.

with examples from 1776.

And for "alternative" the first meaning given (attested from 1540) is

1. Characterized by alternation; alternating, alternate.

Which indicates that the neat distinction Fraser Orr makes simply does not hold in the real world - it does for some people (such as me); but there is ample authority for using the words either way round.