We investigate structured subclasses of Markovian arrival processes (MAPs), focusing on their relationships and modeling properties. Specifically, we examine a subclass of Markov modulated Poisson processes (MMPPs) and a subclass of the Markov switched Poisson process (MSPP) known as Markovian transition counting processes (MTCPs). Our motivation is to understand how MTCPs relate to the widely used MMPP models. A key distinction is that in MTCPs, each increment in the counting process corresponds to a change in the background Markov process, whereas in MMPPs, transitions in the background process are not directly observable. This fundamental difference prompts us to explore methods for approximating MMPPs using MTCPs, leading to the introduction of a hierarchy of subclasses within MAPs and an exploration of their modeling and analytical properties.