People are going to have to start using their own common sense, basically don't use public transport unless your journey is essential.
But the guidance is that public transport can be used when you can't use another form of transport. The lockdown is beginning to be eased tomorrow, to start allowing some non-essential activities again. As yorkie has said many times above, if the rules allow for you to do something by driving there, but you don't have a car, you can use public transport for that purpose.
I assume the exclusive use bit applies to peak hours, obviously some people have to use public transport for shopping, medical appointments etc. I think the message is basically don't use public transport unless you have to, I know it goes against everything that we've been told in the past but such is the unprecedented situation that we're currently in.
There is no exclusive use... I know the line for TfL's services at the moment are "essential journeys only" (basically what you've said above) but this has to change once lockdown rules start being released.
Anyway, from reading comments elsewhere, it seems as if there are quite a few more people venturing out now onto public transport. People need to travel to see others at some point, and to get to work at another point (likely to be much sooner than the former). They're going to use public transport and there won't be anything that can be done about it.
As much as I hate to use this example, remember everyone
has to pay for their train journey! Doesn't stop millions of pounds a year being lost to fare evasion. The point is, you can tell people they can't use it, and that's worked up until now, but people will use it if they want and pretending otherwise is wishful thinking that doesn't match up to reality, because we don't have the manpower to enforce this. And if we did, we'd have a lot more staff hanging around on stations and trains which reduces capacity for other travellers.