I don't call PE and KE, for example, the same thing.
I don't call them the same thing either.
But I call them both energy.
You might say a Mitsubishi 3000GT is not the same
thing as a Dodge Stealth. But they're both cars.
And they have many, many, many essential properties
in common.
-- Sometimes "splitting" is appropriate, making many
distinctions and sub-distinctions.
-- Sometimes "chunking" is appropriate, combining
many things into groups and super-groups.
Is a spin-up neutron different from a spin-down
neutron? Sometimes it's helpful to say yes,
sometimes not.
Is a neutron the same thing as a proton, with
just a difference in isospin? Sometimes it's
appropriate to say yes, sometimes not.
I never said that KE flows. I never said that
PE flows. But I did say that energy flows.
Can you appreciate the difference?
Perhaps this sheds some light on one of the
reasons why the all-too-common "W+Q" formulation
of thermodynamics causes trouble: It's at the
wrong level of chunking. There are lots of
things you can say about E and S that you
would like to say about W and Q but you can't.
(This is not the only source of trouble, but
it contributes to the trouble.)
E=KE+PE is perhaps the world's simplest example of
an emergent property: the whole has a property
that is not a property of any of its parts.
Energy as a whole has a conservation property
that is not separately posessed by KE or PE or
thermal energy or any of the other sub-categories
of energy.
Hofstadter has some amusing things to say
about splitting and chunking and emergent
properties. I recommend his book as being
readable and well thought out.