As of my understanding 'Alloying' is concerned with metallic systems and the w% addition is much larger. for example Steels, Brass, Al-Si etc. In case of alloying there is a completely miscible solid solution where the individual LPs may vary or not ; and a possibility of new diffraction planes may be there(I'm guessing)
Whereas 'Doping' is concerned very much on ceramic and/or non metallic systems with at% addition, which is very small compared to alloying. Like P/Al doped Si, Sr doped BaTiO3, ZrO2 toughened Al2O3, etc. Doping definitely changes the LPs and almost never causes new diffraction planes only shifts it.
The difference between doping and alloying is the following: Doping means a very low addition of a second material (at.% or Wt.%), which is important for instance for changes in conductivity of a semiconductor. For silicon semiconductors, doping concentration may range anywhere from 10^13 cm−3 to 10^18 cm−3, and thus much below 1 at.%. Alloying means a much higher concentration (typically > 1at.%) of a second element as for instance used with two metals or other material combinations to change the properties of the compound.
I think both alloying and doping can change lattice parameter.Even though alloying is for metals... It can also be a form of doping since adding different metals can be form of adding impurity to a system.