All of those are PhD positions.
The concept of a "PhD program" (or programme) does not really exist in the UK in the same way as it does in the USA. PhD degrees in the UK do not involve any coursework or exams -- you start doing research from day one, so there is no mandatory or planned programme of study which all students follow together. This means that PhD positions/studentships/degrees are never usually referred to as PhD programmes.
Every one of the permutations you describe could be called a PhD position -- that's just a generic term which means you will be working towards a doctoral degree in that role. It's not useful or necessary to draw the distinctions which you have -- the funding and duration of the degree will almost certainly be near-identical (around £15k per year tax-free stipend for 3.5 to 4 years, plus London weighting if you're in London, perhaps an extra £2-3k pa) no matter which of the various options you go with. When referring to the funding specifically, the term studentship can be used, e.g. "Your studentship will be paid on the first Monday of every month" or "You have been awarded a studentship for four years of study towards the degree of PhD".
The main distinction between these options is just where the funding for the position is coming from: the supervisor's own grant (perhaps if they have an ERC or similar), the allocated budget from the research council (i.e. the government), funding via a CDT, or the university's own PhD bursaries.
All would be correctly referred to as PhD positions and you would be called a PhD student or post-graduate research student (PGRS) while doing one. The latter term is often favoured by university admin as it's a catch-all for Master's and PhD students.