TLDR: the information you quoted specifies a brand of paint, not a color. Even if it did, it wouldn't be available or wouldn't match. Better off taking a 1" square sample to the store - they can match it.
Here's the problem. Pigments are not stable. Any paint formula is expressed in ratios of pigments. Pigments disappear, because they outlawed a toxic pigment, the company got merged into another company who consolidated pigments, they altered the pigment families to be compatible with another paint chemistry that company also sells, etc.
Formulas are a table of pigment types and amounts (within a particular pigment system). Paint codes like 43-6010 or G3144 require looking in the book to see what the formula is. Likewise those silly little names they give colors, like Plum Orange or Ratatouille.
And even if they still have the formula and pigments for "Firefly", old paint changes color. It changes in the can, on the wall, and even on the drift card in the manufacturer's file. Even if you got the right formula, it might only match under the outlet covers.
This pigment thing is a nightmare. Paint manufacturers don't want to deal with consumers expecting them to match a paint code from years ago. So they all say the same thing: "bring us a sample, we'll match it." And that, they can do. The scanners are very good.
Even within "makes" of paint, there are "models" of paint. For instance Sherwin Williams makes good, better and best interior and exterior latex (we're not even talking glossy vs satin). They make low-VOC formulations. They also make alkyd enamels and 2-part epoxy garage floor paint. In the industrial world, a lot more.
From what you say from your spec sheet, it sounds like it gives only the make and model of paint. That's to assure the homeowner a quality paint was used. Nothing there sounds like a formula, nor a paint code, nor even a silly color name.
Even the "Designer's White" refers to a base, which is meant to be custom tinted at the store. It is saying there's enough white pigment in the can already that you could paint it directly, and will require less custom pigment if your final goal is a soft, near-white pastel. That is not true for the vivid-color bases, which have almost no pigment, to avoid washing out the bold oranges, blues or whatever that will later be added.
They are correct: there's no paint color code there.