In our ongoing study, we explore the concepts of I3-Cauchy and I3-Cauchy for triple sequences in the context of random 2-normed spaces (RTNS). Moreover, we introduce and analyze the notions of I3-convergence, I3-convergence, I3-limit points, and I3-cluster points for random 2-normed triple sequences. Significantly, we establish a notable finding that elucidates the connection between I3-convergence and I3-convergence within the framework of random 2-normed spaces, highlighting their interrelation. Additionally, we provide an illuminating example that demonstrates how I3-convergence in a random 2-normed space might not necessarily imply I3-convergence. Our observations underscore the importance of condition (AP3) when examining summability using ideals. Furthermore, we thoroughly investigate the relationship between the properties (AP) and (AP3), illustrating through an example how the latter represents a less strict condition compared to the former.