There's no official definition for the term "no gluten ingredients" on food labels. However, manufacturers generally use the term when the product does not include any gluten-containing ingredients, but hasn't been tested for gluten or may be at risk for gluten cross contamination.
As of right now, there are no rules governing the use of "gluten-free" on food labels, even though the U.S. Food and Drug Administration proposed gluten-free label rules in 2007.
That leaves manufacturers free to choose how they want to describe their gluten-free products.
Most manufacturers who label foods "gluten-free" make certain those foods contain fewer than 20 parts per million of gluten, which matches the FDA's proposed standards. Companies follow specific protocols, such as testing the raw ingredients and the finished product to make sure it falls below 20ppm.
If a manufacturer doesn't want to test a product, or if that manufacturer is concerned that a product might not consistently meet the 20ppm standard, the company might decide to use the term "no gluten ingredients" instead of "gluten-free." For the consumer, it signals a bit more uncertainty about the gluten-free status of the product.
In fact, several large companies that produce both gluten-free products and gluten-containing products use the term "no gluten ingredients" for products whose ingredients don't include gluten, but which may be at risk for cross contamination or aren't tested for gluten.
Other companies label their products "no gluten ingredients" as a form of legal protection the products may be completely free of gluten, but using the phrase "no gluten ingredients" doesn't promise quite as much.
So should you purchase products labeled "no gluten ingredients"?
Honestly, it depends how sensitive you are and how careful you want to be. Some people can eat food products made on shared lines with gluten-containing products and not get symptoms, while others even need to avoid products made in the same facility.
Before you do buy a product with the "no gluten ingredients" label, take a good look at the ingredients label to see if you spot any allergen disclosures (i.e., a statement like "Made on equipment that also processes wheat"). Companies often will disclose if a product is made on the same lines or in the same facility as wheat-containing products.
If you don't see anything problematic on the label, you'll need to use your own judgment on whether to consume the product. If you're particularly sensitive, you may want to steer clear, or at least sample just a small amount at first to minimize any potential reaction.