There's more than one dialect of English. There's no requirement that they're consistent. Indeed, there's a famous story about a misunderstanding due to speakers of different dialects of English agreeing to "table" an item during a meeting.
Assuming it's the same story, I just happened to read about it last night in Churchill's six volume history of WW2. British members of the Combined Chiefs of Staff produced a proposal which they said they wished to table. Their American counterparts wondered why then had bothered to write it. Once the misunderstanding was resolved, both parties heartily endorsed the proposal.
Back to SIch, it seems clear it differs from paw in that only a body part or implement is used to arrive at its object. I think another difference is what the progressive aspect means. Compare reaching for a book and a train going to a city or a station. As soon as someone begins to reach for a book, regardless of his ultimate success, I'd say he SIchtaH or SIchlI'. But I would only say pawtaH of a train that is entering the city or pulling into the station.
Here are two more things about SIch I'd like clarified.
Can it be used with the body part or implement as its object? ?chabHom bal qoD ghopwIj vISIch I reach my hand into the cookie jar.
Or can the body part or implement be the subject? ?chabHom bal qoD SIch ghopwIj My hand reaches into the cookie jar.
~mIp'av