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We designed and fabricated an epitaxial-graphene-channel field-effect transistor (EG-FET) featured by the asymmetric dual-grating-gate (ADGG) structure working for a current-driven terahertz detector, and experimentally demonstrated a 10-ps order fast response time and a high responsivity of 0.3 mA/W to the 0.95-THz radiation incidence at room temperatures. The ADGG- and the drain-source-bias dependencies of the measured photoresponse showed a clear transition between plasmonic detection under periodic electron density modulation conditions with depleted regions and photothermoelectric detection under highly doped conditions without depleted regions. We identified the photothermoelectric detection that we observed as a new type of unipolar mechanism in which only electrons or holes contribute to rectifying the THz radiation under current-driven conditions. These two detection mechanisms coexist in a certain wide transcendent range of the applied bias voltages. It was also clearly manifested that the temporal photoresponse of the plasmonic and photothermoelectric detection are comparably fast on the order of 10 ps, whereas the maximal photoresponsivity of the photothermoelectric detection is almost twice as high as that of the plasmonic detection under the applied biases conditions. These results suggest that the ADGG-EG-FET THz detector will be promising for use in 6G- and 7G-class high-speed wireless communication systems.