How are "needs to be washed," "needs washing," and the regional variant "needs washed" to be distinguished"?

From a strict descriptivist standpoint, your "the car needs washed" construction is not grammatically incorrect - if we're talking about the Central Pennsylvania flavor of English, that is. But your friend is right that this construction is not found in more standard varieties of English, and can be considered incorrect in formal writing.

Anyway... I think the difference between The car needs to be washed and The car needs washing is the degree of cleanliness that will thus be achieved. The -ing version just says the car needs to get some water applied; the end result could be a clean car, or just a slightly cleaner car. The 'to be' version, on the other hand, states that the desired end result is a completely clean car. Which one is the better equivalent for the "needs washed" construction is something you can answer better than I can.

I will note that in most cases, the difference between the -ing and 'to be' versions of a phrase is pretty subtle, and context is king: if your mom says "your room needs cleaning", you'd better assume she wants a sparkling-clean end result, grammar notwithstanding.

I grew up in the South of Scotland and find this usage entirely ordinary: the construction "your car needs washing", however, to my ears sounds gauche or pompous. Ironically, I immigrated to Ohio which, along with Pennsylvania, I understand, is one of the few parts of the United States within which this usage is commonplace. My son drew my attention to this phenomenon after he moved to Berkeley California and was derided by the locals. He believes this phenomenon followed the Scots-Irish as they emigrated to Appalachia.

They're essentially synonymous. However, if you use the infinitive, you do need the "to".

In everyday speech, the "-ing" variant is probably a bit more common, whereas the variant with the infinitive is probably a little more common in formal usage.

In general where either -ing form or infinitive is possible ("they began doing it" vs "they began to do it"), there's also a slight difference in emphasis between the 'process' vs 'outcome' of an action which may come into play here. So saying "He needs teaching a lesson" focusses a little more on the action of "teaching him a lesson", whereas "He needs to be taught a lesson" emphasises the idea of his new status as a reformed character. But I think it's fair to say that it's a subtle difference and the two expressions are basically synonymous to all intents and purposes.

What no one has mentioned thus far in this thread (I think) is what that construction means and how it's used! Personally, I take it to be similar in meaning to the locution in which a boss says to an employee regarding a particular task: "I'd like it done yesterday!" Similarly, if the grass is really long, making it imperative that it be cut right away, then you use "needs cut" to emphasize that fact. In other words, you are sort of referring to the task as a fait accompli, even before you've done it; that's how soon it needs to be done--I mean, needs done. Make sense?