So the answer is: You only need the one key frame to achieve the desired result. : )
But I looked at your file and here are some tips to help make this work:
After you insert a keyframe on the Offset value 0 of the Follow Path constraint, you might also want to insert a keyframe on the Offset value 100 at another frame in the Timeline. Here it means 100% of the path is traveled, so The difference between the Timeline frames will determine the length of one cycle ( how long it takes for the light to go around the curve once)
Press Numpad 7 before you change the shape of the curve, otherwise you might also move its vertices in the z axis (and change their height), which could change things in an unwanted way.
Before you add the light (and also the curve actually) to the scene, press Shift + C to move the 3D Cursor to the center of the scene. Then the light will be in the will be better aligned with the curve and the "moving in a very small area" will be reduced
Alternatively you can move the light close tho the curve and check the box next to Follow Curve in the Follow Path constraint. This will allow the light move either around or inside the curve depending on the distance between the light and the curve. (If you have the box checked next to Relationship Lines in the Viewport Overlays menu, you can see a dotted line between the light and the curve indicating this distance.)
Hope that helps : )
I also realized the answer to the actual initial question : ) Why does it sometimes look like more keyframes are added? If you extent the panel on the left in the Timeline window (by clicking the tiny ">" icon) it will show a Summary of used keyframes. If you extend this summary, more diamond icons will be shown but they are the same key frames by a different name or UI location. If the summary is extended and the panel is hidden, it can give the illusion more keyframes are added.