I'm using a mining service that supports Segwit.
For the moment, that mining service sends me BTC each mouth on my "1example123" Bitcoin Core Wallet.
Let's say the fork happens and at the moment of the fork I have 0.5 BTC on my "1example123" Bitcoin Core Wallet.
Until I make a decision, I can have :
*0.5 BTC on my "1example123" Bitcoin Core Wallet
or
*0.5 BTC on my "1example123" Bitcoin SegWit Wallet
That mining service will start sending SegWit BTC after the 1st of August on my "1example123" address.
Lets say they send me, in August, 0.1 SegWit BTC.
Until I make a decision, I can :
*End up with 0.5 BTC on my "1example123" Bitcoin Core Wallet + 0.1 BTC on my "1example123" Bitcoin SegWit Wallet
or
*End up with 0.6 BTC on my "1example123" Bitcoin SegWit Wallet
Do I understand it right ?
First of all, this is not Bitcoin Core vs. Segwit. This is Segwit vs. non-Segwit. If the segwit chain is longer than the non-segwit chain, then Bitcoin Core will follow the segwit chain.
To answer your question, you do not understand correctly. You can have both 0.5 BTC on your "1example123" Non-Segwit Wallet and 0.5 BTC on your "1example123" SegWit Wallet. If you are paid 0.1 to "1example123" on the segwit chain, then you can have 0.5 BTC on the non-segwit chain and 0.6 BTC on the segwit chain. There is no need to choose one or the other, you can use both. However using both does take a little bit of work to successfully separate your coins.