Phase diagram of water 
In the phase diagram of water, in relation to the gas phase region and bordering lines, what is the relevant pressure? Is it the partial vapour pressure of water, or is it the total pressure including contributions from other gases? If it is the partial vapour pressure of water that is relevant, then are we at the triple point any time we melt ice in a controlled fashion, even at an external pressure of 1 atm (because both ice and liquid water have a vapour pressure)? Or do things get more complicated with the phase diagram when you are in a mixed system with other gases present?
 A: 
In the phase diagram of water, in relation to the gas phase region and bordering lines, what is the relevant pressure? 

The pressure we are talking about is the pressure on (and inside) the water. Imagine pure water in a can; squeeze the can and you increase the pressure. This is the pressure we are talking about.

Is it the partial vapour pressure of water, or is it the total pressure including contributions from other gases?

Neither nor. If other liquids/gases/substances are present (if you have a mixture), then you basically have another material. Then this phase diagram doesn't apply. Another material has another phase diagram.

[...] do things get more complicated with the phase diagram when you are in a mixed system with other gases present?

Exactly. For instance, try googling the phase diagram for salt water and you will see a very different diagram depending on the amount of salt.
I am guessing your hope was that you could separate a mixture into its constituents and use the phase diagrams of each constituent. But that is unfortunately not the case.
A mixture does not behave as two separate materials that just happen to be located on top of each other. That would ignore the interaction between them, which in general severally interferes with their combined  phase diagram and with their combined properties. That is why for example the tensile strength of a cobber-nickel alloy is larger than the tensile strength of either pure cobber or pure nickel rather than a value in between. 
A: The pressure indicated in the phase diagram is the external pressure.
