yes.. x is the phase usually phase is the measure of time lead or lag of a wave/particle with respect to another measured to an reference(origin).. for mathematical purposes it is measure in terms of and angle.
x is not the phase, x is the variable of the function as you have defined f(x). If you look at the following function: f(x) = Sin (Θ±φ) Where x = (Θ±φ) Θ is the angle which is subdefined in the context of your function (for example, for a phasor Θ = ωt) And φ is the phase.
Just the notation. In all examples you have defined x as the variable, however you need to say the following for the expression to make sense: 1) x = Θ 2) x = ωt 3) x = ωt ± Θ (although in this example you have defined the phase with Θ, which is not considered standard practice, generally use φ or ø) However it is not really correct to define f(x) in this way as you are declaring x as a variable of the function where the function operates in other terms, i.e. Θ, t, ω and φ (this is more my fault for using this notation in my previous reply). More precisely, you should say: v(t) = Vpk Sin (ωt ± φ) Which says the sinusoidal voltage operates as a function of time with angular velocity ω and phase angle φ.