This is a question about comparing results from the Fama french 3 factor model.
I have not physically done this, but let's assume a Fama French 3 factor regression was performed for Coca-Cola (KO) and Pepsi (PEP). The model used was: $$r_{it}-r_{ft1}=\alpha_i+\beta_{im}(r_{mt}-r_{ft2})+\beta_{is}SMB+\beta_{ih}HML$$
$r_{it}$: return of Asset (either KO or PEP)
$r_{ft1}$: Risk Free Rate (3-month T-bill or equivalent investor uses), also the benchmark in this example
$\alpha_i$: what we are solving for aka output from regression aka intercept, Portfolio (Asset) Return minus Benchmark Return
$(r_{mt}-r_{ft2})$:Market Return minus the Risk Free Rate (3-month T-bill or equivalent investor uses)
Assume the rest of the variables are their regular assumptions as found in textbooks
Now I distinguish between $r_{ft1}$ and $r_{ft2}$ because I have read on this site found here that the $r_{ft}$ is the benchmark, the risk-free market return. Now in their original model, they did not distinguish the 1 and 2 on the risk free rate as I did. This leads me to think that $r_{ft1}$ is interchangeable with a benchmark such as the S&P 500 for example.
My question is, this alpha value solved assumes the benchmark is risk-free rate universally across all assets. Although this is a way to compare all assets, that doesn't mean in theory you can compare two different funds/portfolios/assets this way when they are comprised of different items. You should use the other definition of alpha=Asset minus Benchmark return. So can I change the $r_{ft1}$ to be the benchmark of my asset.
Is this an acceptable/practiced method of thinking about $r_{ft1}$?
Lastly, it seems like two different definitions of alpha are being used. Define alpha as Portfolio (Asset) Return minus Benchmark, how we tend to think of alpha. But re-arranging the model would make alpha equal to the Portfolio Return minus Benchmark plus other factors. So now alpha = alpha + stuff. As you can see I am lost and need some clarification about alpha in Fama French.