Yes, yes, yes and yes we are all agreed here.
Is there is a terminology issue here?
DH correctly referred to 'internal energy' here. These days internal energy is normally given the symbol U to distinguish it from other energies. E is an old symbol for internal energy (eg Moore and Moelwyn-Hughes) but is now the general symbol for energy.
Internal energy of a closed system like a thermomenter is clearly not conserved if you heat it up.
One of the substantial sources of error in thermodynamics is failure to define the system appropriately.
It is often really helpful to consider lots of different cases.
In an isolated system U cannot change, but, although necessary, this is not a sufficient condition for the system to be isolated.
This is where I disagree with DH.
The first law tells us that energy and work can flow into and out of a system with a resultant zero net change in U. But only for open or closed systems, since by definition no energy can flow into or out of an isolated system.
However if we set delta U, q and w to zero in the first law it is still not sufficient to define an isolated system since mass may enter or leave the system whilst all terms in the first law are zero.
Would you like to consider examples of how all these cases might be realised?