You can play them as any. The fluff in the book describes them as hailing from the Storm Trooper regiment, but it doesn't take much to play such a character as a Cadian Kasrkin, or as a Grenadier.

GW fluff aside, I thought it was a rather clever way of offering the usual increase in player freedom. You can do an all-ST squad and make a Drop Troop unit with the regiment creation rules (personal recommendation: Schola Progenium, Drop Troops, Iron Discipline, Well-Provisioned … no CO bonus), or you can use them like it says in FFG's fluff (individual ST attached to and moving with a normal squad), or you can play them as one of the many "local variants". It really comes down to what the group prefers.

 Don't forget the term 'regiment' isn't entirely consistent in our own world, let alone in 40k! I think of it like the British Army's 'Royal Regiment of Artillery' which is made up of other regiments and isn't a literal, tactical unit as the term is used in other armies, but more like a corps (though again, not a corps as in two or three divisions, but a corps as in the US Marine Corps).

Hope that helps 

Stormtroopers aren't exclusively from the Schola Progenium, some regiments train their own soldiers as Stormtroopers, such as the Death Korps of Krieg Grenadiers. Wearing heavy carapace armour and wielding Hellguns/Hot-shot Lasguns, they're stormtroopers in all but name.

"Grenadiers are chosen from the ranks of infantry squads, usually by their Watchmasters and recommended for appointment as a grenadier.  This is not a promotion, as grenadiers still carry the same rank, but they are regarded as having a seniority over rank and file guardsmen by dint of their experience and veteran status."

Imperial Armour Volume 5 - The Siege of Vraks, page 88

"The Krieg reminets maintain and train a strong and sizable force of grenadiers, forming an elite force withing the Death Korps. They are drawn from veterans and survivors of decimated squads and platoons. An officer Watchmaster or Commissar selects each grenadier for service."

Imperial Armour Volume 5 - The Siege of Vraks, page 91

Emperor Castaigne said:

Elysians and Harakoni field integral Stormtrooper Company among their ranks. But most other regiments don't have integral Stormtroopers.

Emperor Castaigne said:

If one were to go by GW's material, anyways.

As already mentioned, one could easily just take the existing Storm Trooper career and play it as a Grenadier hailing from that regiment - such a character just would not be a "real" Storm Trooper, but some few regiments do approach the STt's level of quality and equipment … most notably the Cadian Kasrkin (on the tabletop, the only Grenadier type that actually uses the same rules as "true" Storm Troopers, and is specifically noted for being their equals in the Codex: Eye of Terror).