Are there other ways to organize declensions other than the traditional numerical method? If so, what are the pro and cons of that system as compared to the standard system of the first declension, second declension, etc.
It is not an ordering, but it is common in the context of historical linguistics or comparative Indo-European linguistics to categorize nouns by the ending element of the stem:
- first declension is ā-stem nouns*
- second declension is o-stem nouns
- third declension is consonant-stem and i-stem nouns (consonant stems can be divided into t-stems, c-stems etc.)
- fourth declension is u-stem nouns
- fifth declension is ē-stem nouns*
*The treatment of length may vary, as there are not really any noun stems that differ solely in terms of length of the stem-final vowel. The first declension stem vowel was historically long, but was shortened in Latin in many contexts, including the nominative singular form.
An example reference that discusses this kind of categorization: https://www.researchgate.net/publication/272789273_The_phonological_basis_of_Latin_case_patterns_1