From above comments sounds like you're already familiar with some necessary background knowledge, this answer is to further address your concern of the possibility of further classification of procedural memory into motor and cognitive subtypes. Actually from WP source of procedural memory the answer is affirmative:

It took time for the idea of multiple memory systems to become firmly established. In 1962, the severely impaired amnesic patient H. M. was reported to be capable of day-to-day improvement in a hand–eye coordination skill, despite having no memory for the practice sessions (Milner, 1962). Nevertheless, subsequent discussions of memory in general and amnesia in particular tended to set aside motor skill learning and to focus on the unitary nature of the rest of memory. Amnesia was considered to impair memory globally, with the recognition that an exception should be made for motor skills.[16]

Also it's worth mentioning memory could serve as both a static object imagery and a dynamic tool, not as conventionally thought only as some static images to recall:

Memory is treated as an object in recall or recognition; it can be inspected and described to others. In this case, the focus is on the past. However, memory (from the past) can be used as a tool to perceive and interpret present events... when used as a tool, the use of a memory is unconscious because the focus is not on the past, but on the present that is being aided by the past memory. Memory can serve as a tool even when one is unable to recall or recognize the influence of the past memory. This distinction between the two functions of memory set the stage for understanding the role of unconscious (or implicit) memory.[8]

Finally the contemporary widely acknowledged embodied cognitionism thesis of cognitive sciences, sensorimotor systems are seen as fundamentally integrated with cognitive processing:

Many features of cognition are embodied in that they are deeply dependent upon characteristics of the physical body of an agent, such that the agent's beyond-the-brain body plays a significant causal role, or a physically constitutive role, in that agent's cognitive processing. This thesis points out the core idea that an agent's body plays a significant role in shaping different features of cognition, such as perception, attention, memory, reasoning—among others. Likewise, these features of cognition depend on the kind of body an agent has.

This conceptual reframing of cognition as an activity influenced by the body has had significant implications. For instance, the view of cognition inherited by most contemporary cognitive neuroscience is internalist in nature. An agent's behavior along with his capacity to maintain (accurate) representations of the surrounding environment were considered as the product of "powerful brains that can maintain the world models and devise plans".[11] From this perspective, cognizing was conceived as something that an isolated brain did. In contrast, accepting the role the body plays during cognitive processes allows us to account for a more encompassing view of cognition. This shift in perspective within neuroscience suggests that successful behavior in real-world scenarios demands the integration of several sensorimotor and cognitive (as well as affective) capacities of an agent. Thus, cognition emerges in the relationship between an agent and the affordances provided by the environment rather than in the brain alone... previously were thought to be highly abstract, now appear to be leaning towards an embodied approach for their explanation.[12] Wilson also describes at least five main (abstract) categories that combine both sensory and motor skills (or sensorimotor functions). The first three are working memory, episodic memory, and implicit memory; the fourth is mental imagery, and finally, the fifth concerns reasoning and problem—solving.

One researcher goes even further, positing that the multiple opportunities provided by human hands shape people's concepts of the mind.[24] One example is that people often conceive cognitive processes in manual terms, such as 'grasping an idea'.