Ultimately, it doesn't matter.
Different people are going to look at it different ways, and get different results. As long as their projected results work based on the inputs they're using, it's fine.
Savings rate is mostly an academic number anyways.
But the difference in people's philosophies comes down to a difference in questions they're asking. One may wonder "How much money, out of all the money I get, am I saving?" In this case, they'd want to use gross, count retirement contributions towards saving, subtract taxes, yadda yadda. Others may want to know how much of their take home they're not spending. So they'd want to look at net, ignore retirement, etc. It comes down to a different question being asked.
In any case, here's what I do to calculate "savings rate." It's likely very different from what many do.
Only count net income. Ignore taxes and also ignore contributions to retirement accounts. Ignore income from other investments. Ignore income from other sources (gifts, investments, etc.) Subtract expenses from net income. Divide that number by net income to get "savings rate."
That basically gives me the percentage of my net W2 income that I'm saving. That's the number I'm interested in. I don't care how much of my gross I'm saving, or how much I'm saving of birthday money, or how much my taxes are, etc.
The basic question I'm answering is: out of the money I take home from my job, what percent can I keep? Thus it makes sense why my contributions to by 457b aren't counted in my "savings rate" (I don't take that home). Or why my taxes aren't counted, or why I don't count my income from my rentals. I only see how much of my W2 take-home income I can not spend, and invest.
In the end, there is no one definition, because different methods work for different people, and that's fine. :)