Learn English – Meaning of “A man has as many masters as he has vices.”

Phrases.org.uk concurs with the OP that tide referred to a period of time:

The notion of 'tide' being beyond man's control brings up images of the King Canute story. He demonstrated to his courtiers the limits of a king's power by failing to make the sea obey his command. That literal interpretation of 'tide' in 'time and tide' is what is now usually understood, but wasn't what was meant in the original version of the expression. 'Tide' didn't refer to the contemporary meaning of the word, i.e. the rising and falling of the sea, but to a period of time. When this phrase was coined tide meant a season, or a time, or a while. The word is still with us in that sense in 'good tidings', which refers to a good event or occasion and whitsuntide, noontide etc.

Wiktionary confirms this in its etymology for the proverb:

A figura etymologica – time and tide respectively derive from the Germanic *tīma- and *tīði-, which are ultimately related.

A figura etymologica (or etymological figure)

is a rhetorical figure in which words with the same etymological derivation are used adjacently. In the narrowest definition, it is restricted to specialized uses of the accusative with cognate verbs (e.g. live a good life, sing a long song, die a quiet death). In modern linguistics, this same construction goes by the name of "cognate object construction" abbreviated COC. In its less restricted sense, the figura etymologica refers to just about any sort of repetition of cognate words in relatively close proximity to each other.

Its use in rhetoric suggests that it's simply window dressing for additional emphasis.

Related terms: polyptoton, hendiadys.

It's a pun on the word "split", vs. the idiom "to split". In English, a common, informal expression is "having to split" for "having to leave". (This comes from leaving a group of people being effectively the same as splitting off from them.)

The verb split itself, of course, can mean to be literally split into two halves.

In this case, we had a pun because the man (Buzz Saw) was split into halves. Another man asks where Buzz Saw went (implying that Buzz Saw is absent at this point, i.e. has left). Schwarzenegger says that Buzz Saw has left (he had to split). Unbeknownst to the asker, however, Schwarzenegger also literally split Buzz Saw in two.