Trying to understand SegWit
Is my understanding correct please?


The only nit above is the total amount of data is increased a little - so little that it can be ignored.


  Does it mean that for old-fashioned Nodes who don’t look at Witness side-chain, they can only verify transactions up to the SegWit implementation, and will have to reject new blocks because they won't find a valid signature within the new blocks?


Wrong. SegWit outputs look like AnyoneCanSpend outputs to legacy nodes (AnyoneCanSpend = P2SH address that can be spent without a signature - you just have to know the script to spend it.) While new nodes understand the "Witness Program" and verify witness signatures, legacy nodes think it's just a usual AnyoneCanSpend script. Hence, they accept transactions that spend Witness addresses.