What is a lightning network?
There is only one lightning network, not a class of networks called lightning networks. The lightning network is specific design for a network of payment channels. Users open payment channels with other users (only 2 parties per channel) and can route payments to other people through channels that are already open.
Is SegWit a lightning network?
No, nor is it a network of payment channels. Segwit is unrelated to payment channels.
So let me get this straight, SegWit is a proposal to decrease the space that is taken up in block.
No. Segwit does many things, but it does not actually decrease any space. It explicitly will increase the size of blocks.
First of all, segwit defines new output types. These output types put their signatures and any scripts in a new field in the transaction format known as the witness. This fixes the transaction malleability issue. Then segwit redefines the metric for block size maximum to be something called the block weight. The block weight applies a "discount" to the witnesses. The effect of this is as if transactions with witnesses are smaller than their non-witness counterparts, so more transactions can fit in a block (note this is not actually what happens).
Now 90% of the nodes operating in the Bitcoin network accept it and august 1 is the date allotted for a hardfork. SegWit going to be implemented in the Bitcoin network on this date. And because of this implementation, the remaining 10% of nodes will split the blockchain in their own little cryptocurrency or the "old Bitcoin". - Is this correct?
No. The deployment plan for segwit is that if 95% of the blocks in a 2016 block difficulty retarget period signal for segwit, then segwit will become locked in (a 2016 block grace period for everyone to upgrade) and then activate. Segwit is a soft fork so not everyone must upgrade.
On August 1st, BIP 148 (which was not written by the authors of segwit nor is it endorsed by them) will activate. What BIP 148 specifies is that all blocks after August 1st must signal for segwit, otherwise the block is invalid. This is also a soft fork as not all nodes must upgrade. However, all miners must upgrade in order to avoid a chain split. If miners do not upgrade, then there may be a chain split and then there will be two "bitcoins".
Another question: Assuming what I said above is correct, when the chain does split, does 90% of the current Bitcoin marketcap go to the new Bitcoin and the other 10% to the old Bitcoin?
No. What happens to the marketcap and value of the "bitcoins" is completely unknown.
And what is the new Bitcoin going to be called?
That will be decided if a chain split happens and if it persists. There is no plan for a chain split to happen.
