I'm a bit confused here. Do you mean that the cold feed is teed at the bottom of the cylinder and then connected to the PHE inlet?
Yes, either tee the feed for the PHE into the cold feed from header tank to cylinder - or fit an extra flange coupling to have a separate pipe. For the cold end, it's probably OK to just tee them together.
The point is that the water for the PHE needs to come from the cylinder, not the header tank - otherwise there is nowhere for the water to go when you add it back into the cylinder.
I think what would help you would be this :
Print several copies of the diagram, and with whatever writing implement you are comfortable with, draw on the flow for one element - remembering that you cannot create or destroy the water (with one exception).
So for your normal hot water usage (which is the exception) - your flow is from the mains supply, to the header tank, down the feed pipe to the bottom of the cylinder, up the cylinder, to the hot tap, and ultimately down the drain. Between header tank and tap, you still cannot create or remove water volume - so what comes out of the tap must be the same volume as left the header tank.
For your PHE circuit - the flow is from the cylinder, through the PHE and pump, and back to the cylinder. This is a closed loop and the volume of water doesn't change (apart from a minor effect due to expansion). As you had it drawn, your flow came from the header tank, through the PHE and pump, and into the cylinder - but then you have no-where for the water to go other that up the vent pipe which would dump your hottest water up into the header tank.
If you draw these two flows with arrows on the same diagram (use different colours), you'll quickly see that if you tee the feed for the PHE into the cylinder cold feed - the flow can be either way in a short section of pipe, depending on the releative size of the two flows.