Again, your source is a creationist blog post that is based solely on a supposed paper, although the links to the supposed paper point to a non-existent page on another young earth creationist website, likely taken down out of embarrassment.
It’s like pointing to JW.org for the evidence on WTBTS eschatology.
Even so, I’m going to assume they found trace amounts of C14 (as in fractions of percent modern carbon), which if you understand half-life is actually to be expected and at those levels make carbon dating useless as it is almost impossible to accurately measure. But it at the same time disproves the young earth model.
Half-life, the time it takes for an element to lose half its radioactivity and thus this follows a simple formula, 50% you’re at 6000 years (the half in half life) and about the age the Bible would make the earth out to be, 25% you’re now at 20,000 year, more than twice the Biblical age of the Universe, at the edges of accurate carbon dating (50,000 years) you’re at about 1%. At these points the science starts breaking down, you’re starting to have issues with both sample and background contamination - basically anyone who touched the sample or dust from the air or leftovers from previous experiments (you’re blowing up the sample to measure which leaves residue) is introducing fresh carbon.
If you want a young earth, you need everything to have about 50% modern carbon, diamonds don’t have that, your claim is debunked. Again, diamonds aren’t made out of C14, they’re made out of stable isotopes of carbon. C14 is radioactive, C12/13 is not.