We generalize the notions of elliptic pseudoprimes and elliptic Carmichael numbers introduced by Silverman to analogues of Euler-Jacobi and strong pseudoprimes. We investigate the relationships among Euler Elliptic Carmichael numbers , strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I: The former two of these are introduced in this paper, and the latter two of these were introduced by Mazur (1973) and Silverman (2012) respectively. In particular, we expand upon a previous work of Babinkostova et al. by proving a conjecture about the density of certain elliptic Korselt numbers of Type I that are products of anomalous primes. Comment: Revised for publication. 33 pages