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The lows going up the Gulf Stream are not warm-core systems. Just because a low may form over warm SSTs does not mean that it is warm-core; the temperature structure of the storm through the atmosphere, the energetics in play, and the vertical profile of winds all either play a role in determining the storm structure or are markers of the structure.

The Cyclone Phase Space diagrams (http://moe.met.fsu.edu/cyclonephase/) take into account storm-symmetry (as measured in the thickness & height fields) and the vertical profile of winds/wind shear to determine cyclone type, as these factors (primarily the wind profile) include information about the other aspects inherently. Without going into too much detail, a warm core system is going to see its strongest winds at low levels & decreasing with height, while a cold-core system will see the opposite. The CPS mentioned above measures this through thermal wind, or the vertical shear/change in the wind with height, making it an effective measure of classifying storm structure.

Warm-core systems generally develop through latent heat release/transport within the central core, while cold-core systems generally develop in response to strong jet dynamics (e.g. shear) or in the presence of large baroclinicity (e.g. temperature gradients). This latter fact helps explain how many cyclones "bomb" off of the coast of the U.S.: the cold air behind the system & the warm waters of the Gulf stream provide a very strong temperature gradient & ample baroclinicity for such a cold-core cyclone to develop. Hybrid storms lie somewhere between the cold & warm core designations and may incorporate parts of both storm structures/development mechanisms to any number of varying degrees.