I believe you are getting answers, but you might not be willing to accept the answers you have presented with. The people you were raised with and the area you grew up in will differ from the experiences of other humans around this country and others. The etymology of terms evolves over time. Some terms are colloquial (specific to certain areas/regions) others are just what your elders learned based on how they were raised. These terms may or may not be absolutely accurate, but as SushiGaijin, Mezzaluna, and others said above, these terms are common names.
For instance, if you want a Coca Cola, Dr. Pepper, Sprite, Pepsi, etc. you can say the brand name or you can generalize with a common term such as cola, soda, soda pop, pop, etc. All are common names which are indicative to region. The northern part of the US usually calls it something different than the southern part of the US, etc.
People call an item whatever they call it based on who and where they learned it from. It's that simple. It usually is not an issue of whether they are right or wrong when the use is as a common name.
Regarding the searches you are doing, if you read further, you'll find that "pot sets" are often directly related to use such as "stock pot set", "rice pot set", "pasta pot set", etc. All of which require higher sides in order to hold the volume of the contents.
In summary, the answer to your question is that you are referring to common names.
This is similar to the reason why Latin names when used in reference to plants are much preferred over the common names of plants. Why? Because then we know exactly which plant is being referred to. Common names vary according to regions, countries, continents and sometimes even from one garden to another.
"Sorting out Miscanthus sinensis is very challenging. For instance we have seen it referred to as Eulalia grass, Maiden grass, Silver grass, Japanese plume grass, Japanese silver grass and Chinese silver grass. Chasmanthium latifolium is also known as Northern sea oats, Spangle grass, Inland sea oats and Wild oats. See the problem?
Then there is a single common name that is given to two different grasses, such as Bunchgrass (Schizachyrium scoparium and Sporobolus heterolepis). So common names are not very useful to nurseries! Can you imagine ordering bunchgrass, thinking you were getting a tall plant, but ending up with a short one?"
If you use the Latin name, there is no question of what plant is being referred to no matter where you are in the world.
But then... while cookware terminology can get quite specific, it hasn't been standardized by a scientific community so that the terms are universal worldwide.