Yes, there is, at least in one context.
If you are talking about most kind of poetry, alliteration is a rhetoric effect. It does not need an exact definition, because what you are interested in is how effective it is in making the poem achieve something. You can set up rules for yourself, but others might follow different rules, while still thinking that they are using alliteration.
However, if you are talking about the old Norse and Anglosaxon poetry, then alliteration do have a definition, since it is part of the metre. Apart from the structure with stressed and unstressed syllables, lines are held together with alliterations. The rules for this are fairly simple:
- Each consonant sound alliterate with itself, and only itself.
- All vowel sounds alliterate with eachother.
The alliterations can be separated by other words.
I will give an example out of Hávamál, where I've marked the alliterations (the structure is two short lines that alliterate with eachother, then one longer line with an internal alliteration, and then this is repeated):
deyr sjalfr it sama,
ek veit einn,
at aldrei deyr:
dómr um dauðan hvern.
There are more rules for where the words with alliterations should be placed; Snorri's Edda has an entire part dedicated to the different verse forms.
Lars Lönnroth's Den poetiska Eddan has a useful foreword on eddic poetry, which the above is mostly based on. For scaldic poetry, I would suggest a good edition of Snorri's Edda.