But that third type of case, at least as it's applied to East Asian languages such as Japanese, is a result of Western attempts to explain a grammar that's quite different than Indo-European languages. I've encountered several native speakers (of Korean, for example) and learners (of Japanese, for example) who don't agree with the usage of nominative/accusative/etc. for various postpositions (or, if you prefere, particles).
If linguistically, case is considered a syntactic thing as well as a morphological thing, I'm surprised by that. My understanding and everything I've read up to now basically uses case in only a morphological sense for languages that have inflection and use those inflected (case) forms to denote syntactic function more so than sentence word order.
Even the Wikipedia article on grammatical case states that English only uses case for personal pronouns:
Case is a special grammatical category of a noun, pronoun, adjective, participle or numeral whose value reflects the grammatical function performed by that word in a phrase, clause or sentence. In some languages, nouns, pronouns, adjectives, determiners, participles, prepositions, numerals, articles and their modifiers take different inflected forms, depending on their case. As a language evolves, cases can merge (for instance, in Ancient Greek, the locative case merged with the dative case), a phenomenon formally called syncretism.
English has largely lost its case system although personal pronouns still have three cases, which are simplified forms of the nominative, accusative and genitive cases. They are used with personal pronouns: subjective case (I, you, he, she, it, we, they, who, whoever), objective case (me, you, him, her, it, us, them, whom, whomever) and possessive case (my, mine; your, yours; his; her, hers; its; our, ours; their, theirs; whose; whosever). Forms such as I, he and we are used for the subject ("I kicked the ball"), and forms such as me, him and us are used for the object ("John kicked me").
For me, terms like subject/object/etc. are syntactical categories that, although sometimes used colloquially alongside actual case terminology, don't denote the same thing as nominative/accusative/etc.