The links you specified are for a wired (clincher, wire bead) tire and a tube. And the tube will fit a 700x23 tire. I just don't know for sure that you have a clincher rim.
BTW, 700Cx23, 700x23mm, and 23x622 are all designations for the same size tire. It is the third version of a nominally-700cm wheel (A, B, and C
with a bead seat diameter of 622 mm. The tire width is 23mm. Note that the other sizes: 700A, 700B, etc, are not compatible with 700C. If you go by the ISO number, 23x622, you'll guarantee getting the correct size.
As with most things in the bicycling world, Sheldon Brown wrote a good article on tire sizing.
Tire Sizing Systems