The word "fascinator" is derived from the Latin verb fascinate ("to fascinate"), and simply means a thing or person that is enthralling or extremely interesting.
Historically, the term was also applied to a person or animal with the (perhaps magical) power of rendering others unable to move or escape. (wikipedia)
The term "hatinator", which emerged in the early 2010s, is used to describe headgear that combines the features of a hat and a fascinator. (wikipedia) The hatinator is fastened to the head with a range of different connectors, band, comb clip etc just like a fascinator.
The hatinator has the appearance of a hat, while a fascinator does not have a brim.
Headpiece are secured in a range of different ways, combs clips bands cages. what is most relevent to the size and design of the headpiece.