Hello Jaydug,
When dealing with legalities, it really helps if the source document you are using is referenced, so we can at least check its validity and context.
As a general point, the HSE have responsibilities that extend well beyond the fitting of gas installations in domestic properties. In a number of these extended specific situations there is no requirement for the fitter to be registered with GasSafe, hence the HSE comment in brackets. Generally speaking these situations are covered by some other formal process of establishing competencies of fitters. Private self propelled and touring caravans are examples.
A key point the HSE quote highlights is that whoever does the work on any gas system it must be up to the required standards - that is something that caravanners often overlook when decideding to exercise their DIY repairs. either through ignorance of what the standards are or just incompetent workmanship.
I make no bones about the fact I will often deliberately fail to mention the private caravans right to to do DIY on the gas systems, because I know from extensive experience the dangerous mistakes that some owners make. I prefer to promote the safer practice of having a professional job done.
But the GSI&UR are quite specific about the need for ALL gas fitting in domestic properties to be carried out by a fitter who is registered with the organisation currently charged with operating the registration process GasSafe. Simply being "competent" is not enough to satisfy the law when working in domestic properties, you have to registered and carry the ticket for the type of work involved.
That is not at odds with the quote you give from the HSE.
I used to be CORGI registered for LPG, but that lapsed when I retired and on quite a few occasions since friends thinking I'd be competent have asked me if I would do a job for them, and I have to decline because I'm not registered.