If a miner included a Segwit transaction to a block with invalid signature, it would be rejected by Segwit nodes, however not by non-Segwit nodes.

Certainly. It must attain the miner so both despatched to them immediately or propagated to the miner by non-SegWit nodes. SegWit nodes would reject it and never propagate it.

Would the latter take into account the transaction legitimate? If sure, would not this lead to a sequence break up?

Full nodes implementing SegWit guidelines would reject a block with an invalid SegWit transaction in it as would miners implementing SegWit guidelines. What would occur in your situation is the miner would suppose they’ve efficiently mined a block however full nodes implementing SegWit guidelines would reject it and miners implementing SegWit guidelines would refuse to mine blocks on high of it. So it could actually trigger momentary confusion amongst full nodes not implementing SegWit guidelines however the longest chain with most proof of labor would find yourself not together with that block and that’s the chain that even the complete nodes not implementing SegWit guidelines would find yourself following.

Definitions aren’t all the time clear (e.g. re-org, chain break up) however there would not be a lasting chain break up until a subset of miners refused to reject this block that did not meet SegWit guidelines and continued to mine in perpetuity blocks on high of it. This might be a really pricey endeavor as they might successfully be mining an altcoin and never what is taken into account Bitcoin.