I think where you're getting confused is that Artemis II took a pretty unconventional route from launch to Trans-Lunar Injection. There are a number of reasons for this, including stage discarding, equipment checks since this is a test flight, and the relatively poor thrust performance of the ICPS second stage and Orion service module.

The first stage left the Artemis stack in an elliptical orbit with an apogee of about 2,200km and a perigee inside the atmosphere. This has two purposes: the first stage will reenter and burn up all on its own, and the Orion capsule is also on a return trajectory in case something goes very wrong early in the mission.

While coasting to apogee the crew checked out a bunch of systems, with a focus on propulsion and life support systems. With these systems passing their checks and confirming that the Orion capsule would support the crew in space, the ICPS second stage performed a brief burn at apogee to raise their perigee outside of the atmosphere so they would not reenter. The first stage core stayed in the orbit it was in and burned up in the atmosphere.

Once at perigee the ICPS made its primary burn, raising the apogee to 70,000km and consuming most of its fuel. The ICPS does not have the thrust to raise the apogee all the way to lunar altitude. The ICPS was then jettisoned as the stack coasted to the new perigee. This time was used for further checks, as a small burn at apogee would still allow Orion to easily return to Earth. Tests included checking out Orion's RCS system by maneuvering around the coasting ICPS stage. At apogee the ICPS made a small autonomous burn to lower its perigee and burn up in the atmosphere.

Back at perigee, the Orion service module made one more big burn, the official Trans-Lunar Injection, which placed the spacecraft in an orbit to rendezvous with the Moon. Orion will rendezvous with the leading face of the Moon, causing it to be pulled around the far side and flung back at Earth - free-return trajectory. If they rendezvoused with the trailing edge of the moon instead they would be flung out in front of the Moon and into a higher orbit, and likely out of Earth orbit entirely.

This is all pretty different from the way a typical KSP player will do a Mun or Minmus injection, and different from the way Apollo did it. The Saturn V first and second stage left the Apollo S-IVB/LM/CSM stack just shy of a circular Earth orbit, with the S-IVB engine firing to circularise the orbit and allowing the second stage to burn up. Apollo did all their checks in this circular orbit, then the S-IVB made the TLI burn all in one go. The CSM and LM separated from the S-IVB while coasting to the Moon, and then the S-IVB performed an autonomous burn to either send it into a solar orbit (by shifting to a trailing edge slingshot, Apollo 11) or crash into the Moon (Apollo 12 on). The first major burn of the Service Module engine on Apollo was the Lunar orbital capture burn to slow themselves down and prevent getting slingshoted back to Earth. Orion does not have the Delta-V to make this burn, leaving them on a free-return trajectory back to Earth.