As you have observed, the dictionary definitions of 'cranky' and 'grumpy' are virtually identical. As a native British English speaker, my perception was that 'cranky' was more American, although we do sometimes use it in Britain. However, a look at Google books through the ngram viewer surprised me...
This ngram shows that, in British English books only, 'grumpy' is used much more than cranky. But while 'cranky' has been used more than 'grumpy' in US English works, usage of 'grumpy' seems to have overtaken it since 2012.
Based on this evidence, I would say that it really won't matter which you use. If you are writing for an American English audience, you should perhaps consider the age of your audience, or if you are writing prose, the age of your character. Given that this shift in use appears to have taken place in the last decade, an older person is perhaps more likely to use 'cranky'.