Finally getting back to the party. Hope I'm not too late.
Re: First 2 pics here - https://www.flickr.com/photos/43836144@N04/
First pic shows various widths of small wood slats. They are about 1/8" thick. These are used to hold the cane in place, once the cane is pressed into the grooves. This pics also shows a homemade wedged hand tool....
Pic 2 shows the wedge tool's tapered edge/end.
You can fabricate your own simple tools, as I did. Paint stirring sticks, cut in 4" lengths, would suffice for the slats and likely the tool for initially pressing the cane into the grooves. You might break a few thin pressing tools, as you need to press kinda hard to get the cane into the groove properly, i.e., deep enough. You want to press the cane all the way to the bottom of the groove, hence the cane needs to be soft enough to bend that deep, without breaking. Use the thin slats to coax the cane, a little length at a time. Work the cane into the groove a little at a time.
Once a DIYer becomes accustom to caning, these slats may not be needed, as with the pro in the video below.
You will soak your cane in hot water for at least 20 minutes, maybe more, to soften it. The soaking time depends on the cane size.... larger cane requires a little more time soaking. You want to make sure it is soft enough that it doesn't crack or break, when you press it into the grooves. You use a wedge type tool to press the cane into the groove and use the thin slats to hold it in place, while you press the next inch or 2 into the groove. You will repeat this process all around the seat.
This video is a pretty good teaching demo. https://www.youtube.com/watch?v=DdOJfyouk2Q
What I don't agree with, in this video, is, at the beginning (words on the screen) they state to soak the cane and the spline. I don't recommend soaking the spline. If you soften the spline, it will be crushed when you hammer it in. Even in the video, it seems the spline is still hard, so I don't think the guy soaked the spline. Soften spline will cut easily, not requiring a pounding with a sharp chisel, as was shown in the video. They make mention of using a block, to further drive the spline into the groove. A softened spline will crush using the block, as well. For your initial short spline piece (as used/shown in this video), initially holding the cane in position, you may practice with dry spline and softened spline, to see the difference in the two applications. I highly suspect you will crush the softened spline.
Start your caning in the front center, align the weave, then set the back center in place.
With regard to your damaged groove that is too wide, go ahead and set your cane and spline in place. On the outside, where the cane edges flair up, insert some sort of small wedging on the backside of the cane, to push the cane/spline as far to the interior as it will go. Don't cut off the excess cane until it has dried. Once dried, affix/glue, as best you can, some sort of permanent wedging in that small space. Once the glue has dried, then cut the excess cane away. If need be, use small tacks or finishing nails, to hold the spline in best position along that damaged groove area. It's not uncommon to use small nails along the spline, anyway, especially in the corners, for many DIYers and some pros.