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.
The U.S. Food and Drug Administration released final rules for gluten-free labeling in August 2013 (read more here: What Does the FDA's Definition of 'Gluten-Free' Mean for You?). Under the rules, manufacturers who label foods "gluten-free" must make certain those foods contain less than 20 parts per million (ppm) of gluten. In order to make sure foods meet those standards, manufacturers have to test them and follow other manufacturing and ingredient sourcing protocols.
If a manufacturer doesn't want to test a product for gluten, 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 meet the FDA's standards, but using the phrase "no gluten ingredients" doesn't promise as much, just in case.
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. (Read more on this: Should you eat foods made in a shared facility or on shared equipment?)
If you don't see anything problematic on the product 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.