We present a thorough characterization of a red-emitting mode-locked semiconductor disk laser producing the first stable fundamental mode-locked pulse train in the red spectral range. The achievable temporal FWHM pulse length of the Lorentzian-shaped pulses is just below 3 which represents the shortest pulses obtained in a fully-resonant absorber and gain structure design. We also include an intracavity birefringent filter to stabilize and tune the emission wavelength between and . This, in combination with a real-time sampling of the emitted pulse train, allows us to distinguish three regimes according to their round-trip dynamics. The first regime resembles stable fundamental mode locking and is observed for wavelengths below while for an emission wavelength of the continuous wave regime is observed. These two operating regimes enclose the dual-pulse mode locking regime appearing between and . This regime is characterized by an energy exchange between the individual pulses of the dual-pulse mode locking regime and we clearly relate the side-peaks in the radio frequency spectrum to the dual-pulse energy-exchange frequency and the temporal interpulse spacing to the individual arm lengths of the gain-folded external cavity. Overall, we attribute these different operating regimes to near-bandgap operation of the SESAM which allows for dispersion of saturation fluence and modulation depth.