The MPX R1 is an amazing MIDI controller but the description of the relay operation in the users guide is certainly confusing and often generates more questions than it answers. I don't use the relays myself but I've gotten a lot a questions about them and figured it was time to clearify the whole thing once and for all.
By default, the R1 is configured to use its relay "Channels" for toggling relays on and off when most people are simply trying to toggle one of the relays on and off (see the description of Relay Channels below). The fix is to go into the R1's Edit Mode and change the configuration of the 0 or 5 button from Channel mode to "tip" which will toggle the tip on the relay cable.
To make the change, press and hold the EDIT/RUN button on the back of the R1 until you see "Edt" appear on the display followed by the letters "Ch" and the number "1" alternately flashing back and forth. This is the editing mode for the Program parameters. Press the FX button and the display will begin to flash "bYP" and "50". This is the editing mode for the effects parameters which is what you want.
If you press the BANK up button you'll see that the display changes to "1--" followed by a number (the BANK up/down buttons allow you to select which parameter you are editing). The "1--" is refering to button 1. If you keep pressing the button, you'll eventually come to button 5 but instead of a number it will indicate "Ch1". If you press the TAP button you'll notice the "Ch1" will change to "rnG" (the relay "ring"). If you press it again, "tiP" will appear in the display. That is what you want. If you want to change the setting for button 0, keep pressing the BANK button to select that button's parameter and adjust it as well. If you press the button right now, the relay should change for you switching the channels on your amp.
If the amp still isn't changing channels the way you expect it to, there is another parameter in the "Program" Edit mode that can still mess things up. Press the FX button to change to the Program Edit mode and use the BANK buttons to select the parameter parameter "rEL": the "Relay Mode Select". This sets the basic operating mode of the relays. The values are as follows:
1 = Latching
2 - Momentary (normally open)
3 - Momentary (normally closed)
You probably want Latching. Use the TAP and AB buttons to change the value.
You would use the Momentary modes if you want the relay to only stay toggled as long as you have your foot on the button (Lead boost, etc..)
Exit the edit mode by pressing the EDIT/RUN button again.
Relay Channels were invented to deal with guitar amps with more than 2 channels. These amps use a binary coding scheme to define which channel is turned on. Cabling for foot controllers is always a pain in the ass so amp manufacturers do all kinds of tricks to make it simpler to implement. Before I can describe how Relay Channels work, we need to discuss binary numbers.
All computers store numbers as binary data. Binary means that there is only 2 states: on and off, or 1 and 0. In order to get bigger numbers extra positions are added. If you want to represent the number 2 in binary you would use 10. In decimal, the next number after 9 is 10 because 9 is the highest number that can be used. In binary, the highest number that can be used is 1 so counting up would look like this:
These 4 channel amps use binary to select the active channel using the tip and ring of the footswitch cable as the binary digits. Notice that the number 4 in binary uses two digits? One digit is sent through the tip of the cable and the other digit is sent through the ring.
This is a pretty brief description of binary and the MPX R1 relay "Channels" but it should give you a little more insight into what's going on.