What do you call the kind of transfer that takes an orbiting body from orbiting the planet of the system to orbiting a moon of the system. I am merely looking for the name of this kind of transfer orbit (if such a name for it exists). For example, a Hohmann transfer takes an orbiting body from one orbital radius to another orbital radius.
You're looking for injection transfers, which is a type of orbital transfer whereby a spacecraft (or body) is moved from one sphere of influence to another. Specifically, in the Earth-Moon system, this is called a Trans-Lunar Injection.
Injection transfers are not unique to Planet-Moon systems though — for example a Trans-Mars Injection is essentially a Hohmann Transfer between two planets.
Note that this is not the only type of Earth-Moon transfer however: trajectory designs can and have exploited weak boundary effects between the Earth and the Moon called a Low Energy Ballistic Lunar Transfer:
It has been found that low-energy BLTs may be produced by intersecting the stable manifold of an unstable Earth-Moon three-body orbit with the Earth. A spacecraft following such a trajectory is only required to perform a single maneuver, namely, the Trans-Lunar Injection maneuver, in order to complete the transfer. After the Trans-Lunar Injection maneuver, the spacecraft follows an entirely ballistic trajectory that asymptotically approaches and arrives at the target lunar three-body orbit. Because these orbit transfers require no orbit insertion maneuver at the three-body orbit, the transfers may be used to send spacecraft 25 – 40% more massive than spacecraft sent to the same orbits via conventional, direct transfers. From the targeted three-body orbits, the spacecraft may transfer to nearly any region within the Earth-Moon system, including any location on the surface of the Moon.
This effect was first exploited by the Japanese Hiten probe in 1991:
The trajectory Belbruno and Miller developed for Hiten used Weak Stability Boundary Theory and required only a small perturbation to the elliptical swing-by orbit, sufficiently small to be achievable by the spacecraft's thrusters. This course would result in the probe being captured into temporary lunar orbit using zero delta-v (called a ballistic transfer), but required five months instead of the usual three days for a Hohmann transfer. This was the first time a satellite had used low energy transfer to transfer to a moon orbit.
To answer your question narrowly: no, there is no unique name for these specific types of transfers, although a number of transfer types form a subset of what you're describing.
The closest I know of is Trans-Lunar Injection.
A trans-lunar injection (TLI) is a propulsive maneuver used to set a spacecraft on a trajectory that will cause it to arrive at the Moon.
A spacecraft performs TLI to begin a lunar transfer from a low circular parking orbit around Earth. The large TLI burn, usually performed by a chemical rocket engine, increases the spacecraft's velocity, changing its orbit from a circular low Earth orbit to a highly eccentric orbit. As the spacecraft begins coasting on the lunar transfer arc, its trajectory approximates an elliptical orbit about the Earth with an apogee near to the radius of the Moon's orbit.
I'm not aware of any specific term to involve the entire combination of TLI and the capture burn (which is the maneuver performed near the moon, in order to change the flyby trajectory into a lunar orbit).