We consider the experimental trends in a database of specific heat measurements near T-c in high magnetic fields for type-II superconductors with a large value of kappa = lambda/xi, including mostly high-temperature superconductors. Whereas the BCS limiting case is well established in low-T-c superconductors, the exact 3D-XY behavior illustrated by the lambda-transition of He-4 applies only in the particular case of optimally doped YBa2Cu3Ox. Otherwise, transitions are intermediate either between the BCS and the 3D-XY models (e.g., YBa2Cu3O7.00), or between the 3D-XY model and the Bose-Einstein condensation (BEC) (e.g., Bi2Sr2CaCu2O8). The key parameter in ordering this sequence appears to be the product k(F)xi of the Fermi wave number by the coherence length, as evaluated from tunneling spectra in the vortex cores. Such a trend, which is consistent with theoretical descriptions of the strong coupling limit, is visible in the thermodynamics of the phase transition. Implications on the effective mass, the density of pairs just above T-c, the pseudo-gap behavior, etc., are discussed.