Glass and Mirrors - Replacing small pane in puttyless (?) wood door

My son broke one pane of an old wooden small pane door. My examination (sanding the wood lattice piece) suggests that the wood goes all the way to the glass, i.e. there seems to be no putty. I have never dealt with such a thing before. Do I have to remove the pieces of wood that frame the outside of this particular pane? Or must I be deceived and there is really a small section of putty pretending to be wood right next to the glass?

One side should have a removable moulding. Look for finished nail heads.

Thanks. It looks as if you must be right although the nail heads are buried under paint. Looks as if I should take off the three pane long cross molding top and bottom and the one pane long vertical moldings on each side. What a pane pain! But better to know.