Although it looks like an unvented cylinder, with some incorrect plumbing, it could be a vented thermal store with some sort of internal transfer coil for the mains water in (to high pressure hot out). Some thermal stores don't have an external heat exchanger.
The photos are difficult to decipher, but that looks like a vent pipe for that cylinder in middle and the expansion vessel looks like a small (stainless steel?) vessel for the mains supply pipe and coil only.
But I could be wrong, so ask the manufacturer for exact type you have.
The unit might be direct connections for the stove (they can be indirect also) and looks like your plumber altered the return to put a pump on. If the cylinder is designed for connecting a stove, then the stove pipes should have worked solely on gravity (no pump) and should have had an open vent. I assume the pump to the right side at the wall is the outlet pump to rads.
Get another heating engineer who actually knows about solid fuel pipes, as it is clear your plumber hasn't done it right.
Your manufacturer advice might have been correct about the cylinder being suitable for a stove (if it is a vented thermal store), but the statement about the lack of vent on the stove is wrong, as it requires it. Manufacturer might be confused plus inexperienced with pipework from a stove.
Only an idiot would cap that vent and add a pump on the primary return. Would be scary if pump fails!
If it is a vented and direct connections thermal store, then it is an easy set up, - stove gravitates to cylinder and radiator pump works on and off through a thermostat as heat rises and falls, but if it was an unvented unit, then a stove shouldn't be connected at all.
Let us know the manufacturer exact details of that cylinder.