Both would be possible.
der, die, das are related to "that", but they are often used more widely in German and can be used for closer things ("this") as well.
dieser, diese, dieses (dies) are specifically "this, these".
So All diese schönen Pferde would also be possible, but not necessary, for the meaning "All these beautiful horses".
Can 'die' be used as a demonstrative adjective instead of just a definite article?
In English, the two split up into separate words (with “that” related to das and “the” related to der) but in German, all the various forms of that word still do double duty as both demonstrative adjective and definite article (and demonstrative pronoun, for that matter).
When you're learning a different language, it's rarely the case that one word "means" one other word in the other language in the sense that the first word can always be replaced with the other one in all cases where the first language uses that word -- words often have a range of meanings and the equivalent(s) in the other language may not have the same range.
So something like Schöner can also mean handsome, can't it? Why not here? is often a better approach than flat-out saying schöner means handsome.
That said, "handsome horses" sounds like a possible option to me, and I've added it now.
Schöner can also be used in "Was ein schöner Tag" which means "What a beautiful day" schön -> Adjektiv (schön-schöner-am schönsten) english translation pretty -> adjectiv (pretty-prettier-the prettiest)
It is true that schön is used and translated into different words but it is npt the word for handsome
Just to say, I'm from Gloucestershire originally and now live in the North West, and I'm not aware of the usage of the word "handsome" being limited to Cornwall. I use it myself occasionally, for such things as horses, dogs and birds. I might even say "He's a handsome fellow" about a man. (Admittedly I am 58 and may be a bit old school".)