Theoretically, Sharpe should be the average of (compounded) excess returns divided by the volatility of the same. It was designed to measure the risk-reward preferring the risk asset to riskless. So everything should be in excess terms.
Obviously, this was a much bigger issue before the GFC, when nominal interest rates might be as high as 5%. Today with rates at zero and loose change, the distinction makes a lot less of a practical difference. Note also that the volatility of absolute and excess will be almost identical unless rates themselves are highly volatile (which they're not, compared to risk assets).
A lot of practitioners frequently do cut this proverbial corner. And not just for convenience/ease-of-calculation reasons. This would be a mortal sin in academia. But if you're a asset allocator, you can argue that your investors don't care about the same theoretical trade-offs as an economics professor. They simply require absolute returns, and it's those that drive their incentives. Irrespective of the path of short-end rates in the future, if long-dated risky assets are priced for x% absolute, it's the Sharpe Ratio based on those that represents the fairer test of future prospects to meet or fail to meet investors' expectations. Similar arguments about the practical relevance of positive real more than positive excess over cash returns also exist.
There is also a legitimate line of argument that what makes sense for domestic investors looking at only domestic stocks and bonds ceases to work so well when you start to apply the framework to foreign assets (especially Emerging Markets, where the local rate is often not riskless). You can get different results depending on how you choose to handle FX risk. Plus if different investors have a different riskless rate on different base currencies, the same stock can have different Sharpe Ratios to different investors. Theoretically, all of this should come out in the wash (assuming covered interest parity holds, which it must to avoid arb). But then you're adding complexity into what would otherwise be a much simpler assessment. And you're essentially turning the purchase of a foreign stock into a synthetic FX carry trade, which is just needlessly messy.
So the short answer is that the original framework is predicated solely on excess returns. But a lot of market participants don't see the world in quite the same way as the textbooks, so are content to look at this in absolute return terms.
With respect to Sortino, there is no single right answer. It stands to reason that if you look at Sortino to compare, refine and/or improve Sharpe, then it makes sense to use the same set of returns to calculate this as you use for your Sharpe Ratio calculation.
However, the framework itself is defined relative to a target/benchmark/required return that is fully at the investor's discretion. None of target = 0 nominal, target = cash = 0 excess, or target = inflation = 0 real are wrong, per se.