How do we know an event is a gamma-ray burst?

Yes, on the right track where the energy produced in a supernova can indicate if it was a LGRB or SGRB.

An indicative tools stems from the light curves of the remnants, both the afterglow brightness and it's region of explosion can indicatively be traced to how long the event lasted. Large gamma-ray remnants are typically associated with rapid star formation regions and are a more luminous remnant, where short gamma-rays are typically associated with little or no star formation regions and are a less luminous remnant.

It's not an exact science, as reproducing the effects and duplicating results have proven vastly difficult, however using distribution plots can definitively separate the differences and provide a theoretical model to work on.

Some minor information can be found at Wikipedia, however further reading and a more in depth view can be found at Cornel University Library. It is a bit of a read and attempts to explain the physics behind emission through mathematics, I'm not entirely sure whether is will provide the answer you are seeking.