What is the definition of partial pressure used in Raoult's law?

In engineering thermodynamics, assumptions are often made about the ideality of liquid and vapor mixtures. An important quantity in these situations is the partial partial pressure of a species in a vapor mixture. For example, a liquid mixture at thermodynamic equilibrium with its vapor is said to be ideal if it satisfies Raoult's law,

$\displaystyle P_i= x_iP_i^\text{sat}(T) \tag{1}$

where $P_i$ denotes the partial pressure of species $i$ in the vapor phase, $x_i$ the mole fraction of species $i$ in the liquid and $P_i^{\text{sat}}$ denotes the pure species saturation pressure of species $i$ at the current temperature $T$.

My question is, what is the definition of $P_i$ used in eq. (1)? From what I can find in my literature, there seems to exist two distinct definitions of partial pressure, that don't necessarily seem to have to agree. If $y_i$ denotes the mole fraction of species $i$ in the vapor, and $P$ denotes the total pressure, some sources online simply define partial pressure as

$P_i \equiv y_iP \tag{2}$

This makes Dalton's law a definition.

On the other hand, some sources define $P_i$ as that pressure which the vapor would exert if all species other than $i$ were removed, and the volume and temperature were kept constant. Generally, for this definition, $P_i \ne y_iP$.

So my question is, which of these definitions is the one used in Raoult's law?