The neutron star transient and 11 Hz X-ray pulsar IGR J17480-2446, recently discovered in the globular cluster Terzan 5, showed unprecedented bursting activity during its 2010 October-November outburst. We analyzed all X-ray bursts detected with the Rossi X-ray Timing Explorer and find strong evidence that they all have a thermonuclear origin, despite the fact that many do not show the canonical spectral softening along the decay imprinted on type I X-ray bursts by the cooling of the neutron star photosphere. We show that the persistent-to-burst power ratio is fully consistent with the accretion-to-thermonuclear efficiency ratio along the whole outburst, as is typical for type I X-ray bursts. The burst energy, peak luminosity, and daily-averaged spectral profiles all evolve smoothly throughout the outburst, in parallel with the persistent (non-burst) luminosity. We also find that the peak-burst to persistent luminosity ratio determines whether or not cooling is present in the bursts from IGR J17480-2446, and argue that the apparent lack of cooling is due to the "non-cooling" bursts having both a lower peak temperature and a higher non-burst (persistent) emission. We conclude that the detection of cooling along the decay is a sufficient, but not a necessary condition to identify an X-ray burst as thermonuclear. Finally, we compare these findings with X-ray bursts from other rapidly accreting neutron stars.