Is there a term in psychology for when a tool is perceived as an extension of your body? Is there a specific term that describes the phenomenon where you perceive a tool as an extension of your own body?
An example of this would be when a skilled artist or tradesman has so mastered their craft that when they use a paint brush or wrench their mind extends their consciousness to the object and they use the tool unconsciously.
Another example is riding a bicycle.  If you are an experienced cyclist, you are usually not consciously thinking about how you are riding your bicycle.  Your mental awareness has expanded to the bicycle and you respond to riding it like it is another limb of your own body
Short answer
Possible interesting terms are:

distal attribution (externalization)
body transfer illusion (rubber hand illusion)
embodiment

Background
This is a very interesting, yet difficult question. I have come up with a few items that may (or may not) be of interest.
Firstly, there is distal attribution, also referred to as externalization. It is defined by Loomis as:

...[the phenomenon] that most of our perceptual experience, though originating with stimulation of our sense organs, is referred to external space beyond the limits of the sensory organs

and by Auvrey at al. (2006) as:

[the] ability to attribute the cause of our proximal sensory stimulation to an exterior and distinct object

Distal attribution occurs for instance in sensory substitution. Sensory substitution refers to the process of substituting one (dysfunctional) sense with another. A classic example being the guide cane. After training and prolonged use, blind people are reported to experience the information not by haptic feedback perceived in the hands via the cane, but to experience the tactile feedback at the tip of the cane instead. Thus they externalize, or distally attribute the tactile feedback in the hand as originating from the tip of the cane.
Another interesting phenomenon that may relate to your question is the body transfer illusion, or the rubber hand illusion (Ehrsson et al., 2005), from Wikipedia:

The [study participants] saw a lifelike rubber left hand in front of them. The experimenters stroked both the subjects hidden left hand and the visible rubber hand with a paintbrush. The experiment showed that if the two hands were stroked synchronously and in the same direction, the subjects began to experience the rubber hand as their own. When
asked to use their right hand to point to their left hand, most of the time they pointed toward the rubber hand.

In other words, after some accommodation, the study participants attributed sensory input to their real hand to the fake arm in their visual field.
Thirdly, and lastly there is (prosthesis) embodiment (Engdahl et al., 2020), i.e.,

the perception of a prosthesis as part of one’s body

This refers to the phenomenon, akin to sensory substitution but in a physically more invasive context and not exclusively related to sensory perception, that  implanted prosthetics become experienced as a part of the body. It is a more complicated issue than externalization as discussed above, and is especially important during rehab where acceptance is an important part of the process. Embodiment comprises at least two processes, namely:

An object may be implicitly embodied if some of its properties are processed in the same way as the properties of one’s own body [...]. Embodiment may also be experienced more explicitly, as evidenced by affective reactions towards the object, perceptions of ownership, and other subjective feelings.

To conclude with a fourth term used in a more popular sense, it's often said from experienced car drivers that (Urry, 2004):

The car becomes an extension of the driver's body

So extension may also be terminology that might be useful, yet it doesn't really harbor any mechanistical explanation as to the why or how this happens.
References
- Auvrey et al., J Integr Neurosci (2005); 4(4): 505–21
- Ehrsson et al., J Neurosci (2005); 25(45): 10564 –73
- Engdahl et al., Sci Rep (2020); 10: 15471
- Urry, Theory Culture Soc (2004); 21(4-5): 25-39