According to my professor's notes, we have syntactic ergativity when the arguments P and S display the same syntactic behavior. But how is the term "syntactic behavior" defined?
The short answer is that there's no exact answer. There is no clear cross-linguistic definition of what a 'word' is, and therefore no real distinction between syntax and morphology. Indeed, there are calls to unify the study of grammar into morphosyntax instead of using the traditional syntax-morphology distinction. Haspelmath (2011) writes: 'Instead of a subdivision of the grammar of sign combinations into morphology and syntax, we can just work with a unified domain of morphosyntax'.
However, in the context of ergativity, I think 'syntactic behaviour' is easier to tell. You can think of it as behaviour other morphological ergativity, which is usually realised as overt coding on core arguments using case morphemes.
Syntactic behaviour, on the other hand, is typically related to syntactic pivots: in clause linkage, which participant serves to link different clauses together? In a nominative-accusative language like English, the pivot is typically nominative, i.e. A and S can serve as the pivot. In a syntactically ergative-accusative language, the pivot is typically absolutive, i.e. S or P. Compare English and Dyirbal, the classic example of a syntactically ergative langauge:
(1) a. I saw the girl and ran away. (I ran away, not the girl. The pivot is A in the first clause and S in the second.) b. bayi yarra baninyu banggun rdugumbi-rru balgalnganyu ABS man came.NONFUTURE ERG woman-ERG hit.NONFUTURE 'The man came and the woman hit (the man).' (Dixon, 1972) (The woman hit the man, not the other way around. The pivot is S in the first clause and P in the second.)
In English, if we want to use the girl to link up the two clauses, we can passivise the first clause. A similar situation obtains in syntactically ergative languages, which instead use the antipassive, where the A becomes S (and is thus absolutive) and the P becomes oblique. Compare the behaviour of English and Waalubal, a dialect of Bandjalang, below:
(2) a. The girl was seen by me and ran away. (The girl ran away, not me) b. yanga:-nj ŋay gila:, ga:ŋa-li-ya: bulaŋ go-FUT 1sgS THERE take-ANTIPASS-PURP meat 'I will go there to get some meat.' (Crowley, 1978, as cited by Dixon, 2004)
Dixon, R. M. W. (1972). The Dyirbal language of north Queensland (Vol. 9). CUP Archive.
Dixon, R. M. W. (2002). Australian languages: Their nature and development (Vol. 1). Cambridge University Press.
Haspelmath, M. (2011). The indeterminacy of word segmentation and the nature of morphology and syntax. Folia Linguistica, 45(1), 31-80.