Yes, to me
diet=creating a calorie deficit
so anything you're doing to do that is a diet. I'm on a diet even though it's of my own making, doesn't follow any book and I don't count calories. However, I eat less than what I need to maintain my weight.
Now you can get wrapped up in semantics here for sure, since some might not like the word "diet" and immediately associate it with "fad diet". Call it whatever you like but in my mind if you're purposefully (so not talking about due to illness, depression etc) eating below your maintenance calories you're on a diet.
I should also add. I don't think
diet=eating healthy
You can be eating just french fries below your maintenance calories. You'd be on a diet but I don't think anybody would call what you're doing healthy. You can also be on a diet even though you don't need to be or be eating foods that harm you (like if you have allergies...).
Clearly you can be on a diet AND eating healthy. It seems like that is the intention of most people here. You can also start out on a diet and THEN eat healthy (something I've seen on here often too because you get more bang for you buck with healthy foods).
ETA: if it's not obvious from my post you can be eating healthy and NOT be on a diet too and can even gain. Certainly it's harder when you're eating healthier food but definitely still possible