I suspect - and it's a guess - that type three you mention in your first post won't have a cap, for a couple of reasons.

The daily caps have been set by TfL, which seems to have been (and may still be) at odds with the train operating companies about simplifying the ticketing process. As well, if one uses PAYG on NR services only, even with a TfL-issued Oyster card, the fares are the remit of the train operators who may take the view that 'if you want to travel on our trains Sunny Jim, you'll pay what we say and like it.'

It does seem wrong not to cap fares for all PAYG use, especially if the desired end product is (supposedly) seamless travel, although that's not specifically included in Boris's Mayor's Transport Strategy (which does include a plan for 'enhancing rail, including Crossrail...' )