In Australia, this form of writing the letter "r" is virtually non-existent.
When I'm looking to buy a font I want to be able to exclude font's that use this style of "r" ...

... but I have no idea what the term is to describe it?
I have attached an example of the word "stroke" that uses this type of "r"

Thank you for any help - I have been driving myself "nuts" trying to Google this!

I doubt it has a specific name. And, even if it indeed has one, I can’t think of a mean to exclude it from searches, except for looking at it, the same way it isn’t possible to exclude fonts with, for example, single-storey |a| or binocular |g|.

Keep also in mind that some typefaces could offer both versions, one as default, the other as alternate.

In old, handwritten manuscripts, the lowercase r existed in various forms, depending on whether it started or ended a word and the letters to which it connected. The Palmer Method (late 19th/early 20th century) of teaching handwriting in the US taught both styles (as is still the case in many other countries), but when it was supplanted by the Zaner-Bloser method in the 1950s and the D'Nealian method in the 1970s, only the bottom-connecting style of r survived.

My parents were taught the Palmer Method, I was taught Zaner-Bloser.

I’ve always thought of the two styles as top- or middle-connecting and bottom-connecting, but there doesn’t seem to be a formal name for them.

 

Thank you very much for the information ... I really appreciate the time you took to respond.
I'm finding many more script fonts with that "American cursive" r
I may be making more of an issue than it is - but it is rarely used in Australia except for the older generation.
Looks like I will continue to review each font one by one.

Again, thanks for relieving my frustration.

The first time I encountered this type of “r” burnt itself into my memory – it was a “Babar the elephant” book in German. The original French has this letter shape as well, see for example here. It does not appear to be a uniquely American phenomenon.