Wow! Some really great information here, what a great thread. Can I give a newbie summary as I see it, and please correct me where I am wrong.

There are two main camps.
SegWit who want to extend the functionality of Bitcoin, and as part of this extension, they want to split the transaction for storage in two chains (is this correct?) The important section regarding the amount of the transfer and the history link stay in the existing blockchain, but because of the reduction in information, more transactions can be stored in a block. This solves the size problem in the short term, and still allows for an upgrade to 2Mb blocks when traffic volumes have increased in the future. This could become effective fairly quickly.

Double blocks want to double the size of the blocks as a temporary solution, and then increase them again in the future as traffic continues to increase, They believe that this can be implemented immediately. Whilst the code changes could be incorporated quickly, it seems to me that no miner would produce blocks over 1Mb in case the SegWit fork wins, and they lose their income. Thus the 2Mb change will not be effective until the fork dispute is resolved.

If core includes some of the other enhancements with SegWit, then wouldn't that become the popular choice of nodes? Also, if I have understood the problem, then double blocks don't seem to offer any benefit in the short term.