Hmm, it reads to me as if it's inspecting new installations, not repair work.
I think your first move is to ask for the safety certificate that shows a new test after repair which the installation passed, or a formal certificate that signs off the failure with details of work done. I don't know the system, but one or the other of these should apply. The landlord is required to have a certificate showing the gas installation is safe to use.
These must give the name of the engineer and the gas safe number. Check these match the records on the gas safe site. Take a photo of the engineer next time, or if one comes again. The gas safe site has photos of the people registered.
If neither is forthcoming (which is sounds like they might not be), or the engineer details don't match then the landlord is operating illegally. At which point you get the HSE involved. The fact that you are insisting on the certificate may be enough to make him actually get a proper one done.