The chair in the photo is indeed webbed cane. It features the most common (and traditional) installation method - the cane is woven in a sheet and held onto the chair by a spline pressed in a groove. The pane may be on a removable frame but sometimes the frames are integral to the structure of the chair. The pale wood strip around the perimeter of the cane webbing is the spline - there is a groove machined around the frame that the spline is pressed into, holding the webbing in place.
To replace it, you'll need a new piece of webbing and a new spline, and some plain wood glue. The job goes easier if you have an old screwdriver or chisel the same width as the spline (or you can buy a purpose-made tool) to clean out the channel. You'll also want a narrow wood wedge with the tip blunted in order to seat the cane and the new spline.
Start by removing the old cane and spline. This can be helped by heating the spline with a hair dryer. Wet down with hot water if you know it won't harm the chair's finish. Look along the spline to see where the splice is (where the two ends of the reed meet). Using your narrow screwdriver or dull chisel, pry up the end of the spline and pull it out of the groove. Pull out all the remnants of the old webbing and use your screwdriver/chisel to clean any old glue or bits of spline or webbing out of the groove.
Check the groove carefully. If there are bits of old material in it, you will have trouble seating the spline. If the edges have become sharp, it can cut the new webbing.
Soak the new webbing in hot water while you're working on the spline. Soak the new spline too. Once the old spline is out and the chair is clean, place the webbing over the chair (make sure you've got the smoother side up - the cane strips are oriented so that the rough cut sides will all face the same direction) and use your wood wedge to press the webbing down into the groove. First, seat a few inches at the top of the panel. Then, pull the webbing towards the bottom of the panel (enough to flatten it out but not enough to stretch it at all) and seat a few inches at the bottom. Then, do the left and right sides. Once the webbing is pressed into the groove in 4 places, it will essentially hold itself in position - you can work around the perimeter with your wedge and press it in to the groove.
With the webbing pressed into place around the whole perimeter, run a small bead of wood glue around the groove - you only need enough to tack the spline in place, the strength comes from the friction fit of the material. Then, set the spline into the groove - when you're all the way around the perimeter, trim it to length. With the spline placed into the groove, use a mallet in combination with the wood wedge to drive it home - you want the spline to bottom out in the groove. The action of driving the spline to the bottom of the groove is what will pull the webbing taut - this step is where the tension in the panel comes from.
With the spline fully seated, trim off the excess webbing along the spline (can be done by running a razor knife along the spline), wipe up any glue squeeze out with a damp cloth, and you're done. Be sure to let the webbing dry fully before putting the chair back into use - if you apply pressure to the still-damp panel, it will stretch and become loose.
Although the process sounds complicated, once you've replaced a few panels it becomes second nature. If your first panel takes you an hour, your third or fourth might only take 10 or 15 minutes.