Getting a buckskin/dunskin
They are correct. There is a test for dun, and for Agouti which your mare may also carry and not express (due to lack of a black gene to throw to points). I looked at your pics and she looks chestnut/sorrel to me. Either way, she has 2 red genes and if you want a buckskin, the dun part is irrelevant.
If you want a GUARANTEE of buckskin, a perlino (not a cremello, who carries no black gene) will give you a guarantee of cream and if your mare carries agouti (only $25 to test for it with mane hairs with animalgenetics.com) you have a great chance at a buckskin without having to have the perlino homozygous for Agouti or black but if you want a guaranteed buckskin foal, yes, a perlino that is homozygous black and agouti is the only way to get a guaranteed buckskin (unless your mare is homozygous for Agouti). Takes one black and one agouti and one cream gene to make a buckskin. Your mare might be homozygous for Agouti...I have a palomino mare that is. All I have to do is breed her to my homozygous black stud and I will get a buckskin every time, guaranteed. She is for sale BTW or buy a foal in utero from her. I have a buckskin/white paint yearling colt for sale too. lewisranch.com