Both psychologists and psychiatrists (and, indeed, other professionals including non-clinicians, e.g. academics, with appropriate experience and training) can provide specific diagnoses according to both the ICD and DSM frameworks. This is explicitly stated in the DSM. Indeed, there are a number of DSM diagnoses that it would be very difficult for a medical professional to give on their own: Specific Learning Disorders and Intellectual Disability being examples that require formal psychometric assessments that medics are not generally qualified to give. There is formal guidance on this issue published by the British Psychological Society here: Providing-diagnoses
Firstly, there is no singly defined list of 'proper' diagnoses. In fact, there cannot be. This is because a diagnosis is a way of describing and categorising a particular sort of problem - and we are finding new sorts of problems or refining the way we understand them all the time. At one time, there was no diagnosis relating to HIV or AIDS - but that's because it either hadn't come into existence or we hadn't yet realised what it was.
With psychological disorders, we have two main systems - the DSM and the ICD. But they are not (and cannot be) somehow the sole source of information for diagnostic categories. For example, there are specific manuals for diagnosing sleep disorders or disorders in young children that encompass things not fully delineated in the main manuals.
There are then all sorts of diagnoses that really only exist because it has been determined that such a condition can occur and people have researched and written about them. In fact, this happens quite frequently with neuropsychological disorders because injury to a specific area of the brain can have effects that just haven't been seen before.
There is perhaps something of a distinction to be made here that can help to resolve any problems. If a diagnosis is made at a high level (e.g. traumatic brain injury resulting in particular levels of functional problems) then the specifics can be provided and related to a particular condition - which might mean specifying the sort of brain injury and the way in which it causes impairment (e.g. thinking you have never seen something familiar before - called jamais vue).
With neurodevelopmental disorders, this is often the required approach. It can be entirely clear that something about a child's brain development is causing them real problems without it necessarily being something that fits the usual diagnostic categories. That doesn’t mean they don't get a diagnosis - it just means the diagnosis has to be given at a high level (e.g. an other specified neurodevelopmental disorder) with as much additional descriptive or explanatory information as is available.
