The clip is an "E" configuration. There are different size tools so don't get one for a GM car.
The idea is, like you said, along the handle, slide the tool in and catch BOTH ends of the 'E' clip. As you push, the clip will expand. It might even come off, which is ok.
Trouble comes when you have only one side of the clip off because the remaining end will still hold the handle on the spline.
If you have a spare (or new) handle, put the tool up to the clip and notice how it works. Some handles have a plastic 'washer' between the door panel and handle. This saves the door vinyl but you need to insert the tool between the plastic washer and chrome handle.
for a YouTube video.