There are no accepted meanings to these terms. They are spewed about like fizz from a shaken soft drink bottle; but, the spewers never clearly define them. Rather, individuals make assumptions of meanings. Certainly, a coin can be made for a special occasion, but there MUST be a relationship between the coin and the event, AND the TPG must publicly establish that relationship. Then, and only then, does "Special Strike" have meaning in relation to that coin, and that coin ONLY.
The term "Specimen" is especially vague, again because those who stick this on a slab label fail to state what that means. If it means the coins looks different, then the label make MUST state exactly why and provide documents to support that. Strangely, few seem to understand that there is a considerable range in appearance of normal coins. These are the result of changes in die wear or damage, die maintenance ('proof-like' is one of the most obvious), planchet finish and upsetting, and even press set-up and adjustment. (See the bologna about "1964 special mint set coins" - they are simply early strikes off new dies.)
Sure, if some want to open their wallets and close their minds to being conned, that is there prerogative. But I'm confident that some people remain with a little common sense and the ability to think through a situation using logic and facts.