syllables are split at the vowel
Seems modern academic currents indeed say it. But it seems to be of limited usage for practical task of splitting words at line ends.
https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D0%BE%D0%B3 says "Понятие слога лежит в основе правил переноса слов в русском языке. Однако, разбиение слова для переноса не всегда совпадает с разбиением слова на слоги, в связи с чем некоторые источники разделяют понятия фонетического слога и слога для переноса". So what I describe below, and what I remember from USSR school, is maybe in modern currents called "syllables-for-splitting" while ones you describe in your question are perhaps "syllables-for-pronunciation". In practice, I can't say why anyone would need latter in the casual life. The former ones seem to be those that practically matter. And the former I would discuss further on.
Слог - syllable - (at least the "syllable-for-splitting") consists of
- initial single consonant (sometimes missing: Russian has no diphthongs, not in writing at least, so two adjacent vowels spawn two syllables)
- a single vowel (the syllable nucleus)
- a number (from zero to many) final consonants
A mid-word syllable terminates at the start of the next syllable, which means at the initial consonant of that.
So, in your case,
Щип-цы, the second syllable is built around "ы" vowel and contains the preceding consonant "ц". That makes the first syllable end at п - the sound preceeding next syllable's ц. That makes the between-syllables dash be inserted between п and ц, not earlier or later.
Granted in practice there may be more rules for splitting words, less strict and giving you more options. http://orthographia.ru/orfografia.php?sid=97
However if you stick with the between syllables borderlines (the safest rule of thumb) then the above holds: syllables include only ONE consonant preceding its vowel (if such consonant exists), not two consonants or more.
Notice, that the first syllable of the word might contain multiple left-side consonants.
Яб-ло-ко has б and л in different syllables, but
бло-ка-да has them both in the first one. Because then there is no "zeroth" vowel-less syllable to own "б".