No; a partial function is one that isn't well-defined for the whole of its domain. So, as per the article, head is a partial function because its type signature of [a] -> a implies that all arrays have a head value. But head [] does not. It's a partial function.
The type system isn't helping you at all there.
The feeling I had was that much of the prelude stuff was there to provide for beautiful, terse examples of functional programming and less to protect a software engineer.
I've just started my Haskell journey and the undefined paths through partial function implementation caught me by surprise.
Just skimming Wikipedia it looks like I would want to use "total/strong functional programming" but apparently "total functional programming is not Turing-complete"
https://en.wikipedia.org/wiki/Total_functional_programming
Also found this on the Haskell site after some more googling:
It isn't, but usually Turing completeness isn't what you want. Take a look at Idris, where functions may explicitly be total or not-necessarily-total.