That kit looks same as the on I got, did my job diagnosing thermostat pipe…if only my kid bro would remember to bring it back.
If your white smoke goes away after the car is up to temp it could just be warmup condensation, both my prince engines do it, and they can look a little plumey these colder nights. The coolant on top of your gearbox might be your thermostat housing and the air lock occurring as it leaked out and being replaced with air due to the potential fracture height in the pipe. You didn’t overheat (can’t recall reading it), hopefully It is just a ‘simple’ leak. When you searching around have a really good nose toward the water pump and a wee touch of renolit or rubber grease can be useful.
If the oil cooler seals have hardened enough to leak it is more commonly oil to the outside noticed first, sort of up behind the cat, if they have hardened enough for oil and water to exchange you should see something as a tell tale, oily combustion gunk in the expansion tank, or cremolla foam in the oil cap and maybe an increasing oil level. For coolant to be entering the combustion zones you have only really got the head gasket and turbo, and if it was the turbo you’d know it was an issue in other ways. If you still need to reassure yourself you could put a camera in through a spark plug hole and see if the piston tops have been washed. Your coolant system pressure test should help ease your mind, and hopefully only identify a pipe.
Thanks appreciate your message, the advice and support. This evenings update.... I don't have a garage so Just went to turn the car around so the front end would be out of the rain under a tarp tied to house and trees

, with weather as it is wanted to clear the n/s mirror, .sat in the drivers seat, im still getting used to dimensions, shallow windows and less than ideal rear view when trying to reverse in dark with rain falling heavy... press the button to lower the window and nothing moves...try the passenger door switch same... the n/s electric window is now non operational

It worked perfectly a few days ago,...can hear the sound of what I presume is the motor engage, or a relay and then cut out, doesn't event show an attempt to move ... ,My list of jobs is growing

Guessing I wanted a project and this is going to be one...Is this French electrics doing what they do, a known issue or I'm just having a run of luck ?
The window can wait.... As far as I know its not overheated...nagging doubts , wondering if it had prior to me buying it ... and that was reason for sale .... Im only just over two weeks into ownership... did the basic checks, ran well and diagnostics were all clean ..everything worked ...heart ruled head im thinking ..., it looks amazing and drove well. and I keep reminding myself everything worked...... . re checked diagnostics just now - nothing logged.. still no EML or warnings on dash.
I've had a good look around the oil cooler and whilst its not clean, theres no significant oil leaking or coolant visible. There is a film of residual oil up near the rocker cover joint both timing chain side and along the back between the manifold but nothing major. Dipstick level looks in the middle , and no mayo ... Frustratingly cannot see any water leaking or dripping, nor hear anything hissing or see visible steam in the engine bay... the coolant in the tank is still pink , up to max and as should be , did another HG test and fluid remained blue.... level hasnt dropped for third day... so adding to the mystery and cause of the significant steam / white smoke from exhaust, no heat in cabin ..... air lock still and coolant being forced into cylinders ? Have had no experience of any car I have owned doing anything similar to this .... it continues even when its reached 90 on the temp gauge . temp gauge doesn't rise.... Because I'm still unsure why, im reluctant to take it out on the road. Cant see any water leaks or even staining on the water pump side.
The thermostat and pressure test kit should arrive so will have to wait for them.... Napa stat unit from Ebay Car Parts in Motion £63 ish
Coolant Thermostat fits PEUGEOT RCZ 1.6 10 to 15 NAPA 1336CC 9804315380 Quality | eBay UK and the kit from Amazon.