Uff, it depends.
The easiest way to look at it is:
for i) difference training, you need FI - a FlightInstructor of any kind and get scribbles in your personal flight book;
for ii) class rating & co it depends, usually you need FE - a FlightExaminer, not necessarily connected to an ATO, and this one can sign into your license with a notice letter to CAA. You can exchange FE by FI-at-an-ATO for such case, but then you have to get a new license from authorities, as under EASA FI is no longer allowed to scribble in your license.
In both cases the plane can be any having the trained specs, BUT - the usual problem is insurance coverage of the plane, not certification: if you take an FE to train something you don't have the license to do, the FE is PIC and the insurance of the plane has to cover that. I remember the CVFR additional training to convert ICAO to PartFCL, where this was a problem due to the fact that most owners had only limited liability on their planes.