That quote about them being "widely different things" just tells me that your source doesn't know what they're talking about (I say that as someone who will be teaching this in college-level biochemistry in about a month). Sure, in a distilled water situation, adding lactic acid would be different than adding a lactate salt, but, under physiological conditions there is no functional difference (and you're not adding a lactate salt, you're adding protons and a lactate, a.k.a. lactic acid).
The protons got added to the solution back when glucose got oxidized. Whether they (2 lactic acids get made for each glucose oxidized, so we're really talking about 2 protons per glucose) briefly had the protons back associated with the electrons donated from the NADH and immediately give them up again or whether they never came back is irrelevant.
The 2 protons are in solution, so if you want to you could call glucose the acid (or NADH+H+) and lactate the base. It doesn't matter, there are 2 lactates and 2 protons created from glucose. Whether you say they were ever 2 lactic acids or not is pointless, if you balance the equation, you made the equivalents of 2 lactic acids from 1 glucose.
Call it whatever you like, it's the same thing and has the same effect on the cellular pH. Either way it's a fermentative waste product created by reducing pyruvate (or pyruvic acid, if you prefer - again no functional difference under physiological conditions) that is generated because you don't have enough oxygen in your cells to continue oxidized the pyruvate all the way to CO2 and your cells need to regenerate the NAD or they'll die.
Whatever you call it, it's exported to the bloodstream and either used to feed other cells or transported to the liver where it gets converted back into glucose (after which point it goes back into the bloodstream).
Edit: Sorry, by the time I got done writing my reponse, I see Polaris did it much more succintly.