Using Monte Carlo simulation, we analyze phase transitions of two antiferromagnetic (AFM) triangular Blume-Capel (BC) models with AFM interactions between third-nearest neighbors. One model has hard-core exclusions between the nearest-neighbor (1NN) particles (3NN1 model) and the other has them between the nearest-neighbor and next-nearest-neighbor particles (3NN12 model). Finite-size scaling analysis reveals that in these models, the transition from the paramagnetic to long-range order (LRO) AFM phase is either of the first order or goes through an intermediate phase which might be attributed to the Berezinskii-Kosterlitz-Thouless (BKT) type. The properties of the low-temperature phase transition to the AFM phase of the 1NN, 3NN1, and 3NN12 models are found to be very similar for almost all values of a normalized single-ion anisotropy parameter, 0<δ<1.5. Higher temperature behavior of the 3NN12 and 3NN1 models is rather different from that of the 1NN model. Three phase transitions are observed for the 3NN12 model: from the paramagnetic phase to the phase with domains of the LRO AFM phase at Tc, from this structure to the diluted frustrated BKT-type phase at T2, and from the frustrated phase to the AFM LRO phase at T1. For the 3NN12 model, Tc>T2>T1 at 0<δ<1.15 (range I), Tc≈T2>T1 at 1.15<δ<1.3 (range II), and Tc=T2=T1 at 1.3<δ<1.5 (range III). For the 3NN1 model, Tc≈T2>T1 at 0<δ<1.2 (range II) and Tc=T2=T1 at 1.2<δ<1.5 (range III). There is only one first-order phase transition in range III. The transition at Tc is of the first order in range II and either of a weak first order or a second order in range I.