Keep in mind, I did my research by looking up each Japanese kick name on Youtube and watching one of the top videos in the results.
And I know "karate" isn't one monolithic thing, so it's possible that a kick may be done one way in your style of karate and another in the video I watched. But from what I saw, I think we have a lot of the same kicks, just sometimes called or categorized in different ways.
Like, I would consider "front kick" to include both the "snapping" front kick with the ball of the foot and also a "groin front kick" which is where you kick upwards and hit with the instep, like one of your groin kicks. And I would consider there to be three types of ax kick - a "straight ax" that goes just straight up and down in the same spot, and an "inside ax" and "outside ax" which do make a crescent shape on the way up (to get around the target) and then come straight down. ETA: Which then is different from a crescent kick, which hits with the side of the foot, while your leg is around the top of the crescent shape.
And then, yeah, the kicks can generally all be done with the back leg, front leg, jumping, turning, etc too.