Your IP address is assigned to you by your ISP dynamically (unless you have a static address, in which case that information will be stored in your router). Dynamic addresses are not stored in your router. It is quite possible that even though you have a dynamic IP that you appear to have a static one. It just depends on how your ISP allocates its IPs.
The router reset button is usually used when your router gets itself into a state that even a power cycle cannot recover from. It resets all of the router settings to the default factory state. You will need to re-enter all information that was previously stored in order to be able to access the Internet. Check with your ISP for the settings to use, and the router manual for details on how to enter it. Most routers allow you to store all settings and save the data to a file which you can place on a USB stick or other storage and from which you can quickly restore if you need to. I strongly recommend that you create such a file, if you haven't already done so. Again, see the router manual for details.