You are just getting caught up in semantics. Saying there is no net force is the same thing as saying that the net force is $0$. Think about your bank account. If you have $0 in it, could you not also say you have no money in it?
With that being said, it's pretty standard to say no net force when the net force is $0$. People will most likely get annoyed with you if you hit them with "well technically there is a net force, but it is $0$." This is similar to if you went up to someone who has no money in their bank account and said to them "Hey don't worry. You technically do have money in your bank account. It's $0 worth of money."
Another issue you can see with what you propose is that the term "net force" becomes completely pointless by itself because you can say there is always a net force acting on an object. This then means you always have to specify "zero net force" and "non-zero net force". Now this technically isn't wrong, but I would say it's pointless. For a point particle, there is no way to distinguish between no forces acting on it at all and the forces acting on it cancelling out, so it makes sense to described these scenarios in the same way by just saying "no net force".
Using your reasoning you could also say that there are always forces acting on something, even if there isn't a force. For an extreme example, as I type this now I could say you are punching me in the head with a force of $0\ \rm N$. I guess you could argue that this is true, but I'm not sure why you would want to.
So, in summary, what you are proposing isn't technically wrong (if others agree on the use of your words), but it is unintuitive and not the common usage of the term "net force".