Can Core see the transaction made to Electrum with 0.5BTC? Restarting Core with -rescan option?
Strangely, no. After I swept keys to Electrum (while keeping Bitcoin Core open), the only transaction that appeared in my history was a small amount (~0.0001 BTC). Otherwise, the bulk of the "available balance" remained unchanged, ie ~12BTC. So yes, it seems a -rescan in order to see how that affects the amount.
For future recovery: your half a Bitcoin has
about $300 worth of Forkcoins. Don't bother about them now, and don't try to recover them before securely moving out all your Bitcoins, but don't forget about them either.
Thanks for this reminder!