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Okay, so I think that the main overarching point of your argument hinges on the fact that art is primarily an expression of a human, created to convey a message, evoke a feeling, etc. While I agree with this point, I think that your application of it to image generation is mistaken.
The first main thing that sticks out to me is you saying "There’s no mind, or creative intent behind a DALL-E image. It’s a rather mechanical process by which a computer program translates a string of text and some random noise into an image.". And, to some extent this could be true if AI algorithms were used to generate content without any human input, at all. However, the resulting images are a work of the person generating them via dialing in all the settings and writing a prompt. Their inputs are passed through an algorithm that returns a new result. Since humans have long since used tools to transform one medium into another, why would using an AI for that purpose suddenly prevent the users from being artistic? To offer a comparison - taking a photograph is a function where the input is a person specifying a camera's location, rotation, optical and digital processing settings - and the output is an image. What specific quality would separate a photographer using a camera to create art vs a human using an image generation algorithm to create art?
To address your other point,

Compelling images are not necessary art, if they do not evoke an emotional response. For instance, we would not classify a company’s logo as art, because it is lacking artistic intent.

While I agree that purely naturally occurring or random things are generally not art unless framed by a human, the example you picked doesn't relate to the point. The job of graphic designers is to use learned conventions in typography, visual styles, color theory and other fields to craft a logo or a brand that invokes certain feelings or memories for the viewer. While it's a highly "corporatized" example, I don't see a reason to exclude graphic designers from the artist designation.
That brief ramble of mine aside, can you prove that AI use will never constitute artistic intent? For example, generation algorithms let humans combine depictions of things with visual styles or other situations that were never seen before, creating brand new depictions. How could this not be a possible venue of artistic expression? And that is just one example that barely scratches the surface.
I have a felling that you are combining two points into one here:
1). "AI art" is not liguistically precise, so we should use "AI image generator" instead
AND
2). Artificially generated art should not be consider "art".
When it comes to 1) - well, it's probably true that "AI image generator" is more precise than "AI art", it is still not precise at all. As an AI engineer I know that the term "artifical intelligence" has nothing to do with any intelligence - it's just really sophiscticated math. So shouldn't we try to force the name "sophiscticated math applied as an image generator" instead? It's much more precise (but probably still not precise enough).
Also - for people who doesn't know what AI is, I think this is a pretty accurate name actually. Images, paintings, etc. are considered art. They are generated by some magic thing called "AI". And we have a pretty descriptive name "AI Art" - is languange more about being scientifically precise or rather about being descriptive?
The 2). pretty much comes down to a very subjective and individual question of "what is art?". And I believe that other commenters had made a clear argumnets for why the whole process of designing and creating such AI generated images CAN be considered "art" and why for some people AI generated images CAN be viewed as "art". But it's purely subjective at the end of the day.
My point is - languange is not about being precise and art is what you consider it to be. It's impossible to have a precise name for everything, especially something as not-easily-definable as a genre of art.
And while I think I may agree with your last sentence

"AI image generator” is more precise and doesn’t lose anything except unjustified hype

it's not a good reason for me to enforce thousands of people to use different name. >

AI researcher here with many published articles at top venues (the reason I write this is because you wrote that you are an AI engineer so we play the same game). And I agree with everything you wrote except:
"that AI does not have anything to do with intelligence"
The truth is more closer to the following: "We have no freaking idea what intelligence is and/or how to define it." There are certain definitions and approximations that we can use, but they all seem relatively blunt (e.g., IQ scores, Turing test), and always fail to convey the whole picture. 
Consequently, that part of your claim is nonsensical. (For all we know, AI as we know it is intelligent to some degree and is on the path towards intelligence.) >

What I mean here is strictly related to my engineering view on AI - that under each method, model, layer, algorithm, etc. we have a pure math and no human-like mind (like I believe many people see it and media like todescribe it). Wheter or not to consider it "intelligent" is to me more like a philosophical question rather an engineering one.
And that's probably a totally different discussion on whether we can consider AI intelligent or not, is it close to human mind or not, etc. and I must agree with you that it is strongly related to the very blurry definition (or more the lack of clear definiton) of intelligence. 
But I don't think it changes anything in my point - in fact I belive that with that side discussion we are only confirming that we can never make a precise enough name for something. >

What do you mean with a human-like mind? Your first paragraph seems highly subjective and vague.
I agree with your two last paragraphs. >

By a "human-like mind" I mean how generally people understand AI - something that makes conclusions out of thin air in opposition to a construct of math and algorithms working on data, a magic box instead of a product of engineering/research work.
And I think you are right- it is subjective. And you might have a valid point here - maybe what we build is actually something similar to human-like mind. Maybe some super multi-layer deep neural network may be similar to human mind in terms of neurons structure in our brains. Maybe our brain is also only a sophisticated math machine. But again - my subjective opinion here is that it's purely philosophical question.  
EDIT: Adding one more thought.
What I am trying to say that for me AI is about solving math equations or figuring out engineering problems rather than shaping something that can be considered intelligence. >

Then we are in agreement. So, it seems I changed your mind, and if you were to rewrite your text you would say something along the lines of "my intuition or feeling tells me that what we have in AI is not intelligence". My feeling tells me that many (if not most) AI researchers would disagree with you. >

Glad we found a common ground - it was interesting discussion!
Yes - it’s a good rephrase of my text there. You are right - I used too strong words for an opinion.
Actually I know plenty of AI people who share similar views to mine. I even met few professors at my uni who openly were against using the word “intelligence” in term “AI” and preferred using “machine learning” as an umbrella term instead (which again is causing other nomenclature problems). >

I meet professors who are some of the most cited on a daily basis. They have different opinions to each other. Most of them don't care about what intelligence is and see it as a crappy word. But whatever intelligence or creativity is, they would also agree that it is clear that systems like AlphaGo and GPT3 exhibit some part of it.
In regards to ML: It is a part of AI. Any professor or researcher who does not know this is a professor you should ignore. The first AI systems were based on logic and planning, which use no ML at all. For example, multi-agent systems use e.g., game theory and optimization to a great degree, and have outperformed ML systems in many applications. >


Compelling images are not necessary art, if they do not evoke an emotional response. For instance, we would not classify a company’s logo as art, because it is lacking artistic intent.

Boy, I'm sure the people who make them disagree. A lot of effort and intent goes into those.
And say, what about the logo of an art company?

Things which are beautiful and evoke emotion are not art if they lack artistic intent. For instance, a sunset, or a rock formation, or a plant which grows in a fractal pattern may be beautiful and affect the mind in an art-like way, but we don’t call it art because it’s essentially just a result of math and physics and not human expression. AI images are actually a lot like that.

Math is very much used for artistic purposes. From things like the golden ratio to mathematical generation, like the spirograph >

If we're talking about effort and intent as an argument for why a company logo could be art, it's important to note that AI art lacks both effort and intent. >

AI doesn't generate anything without somebody telling it what to generate. The intent comes from there. >

Yes, there's intent on the part of the person inputting a text prompt, but that's analogous to the intent of the wealthy patrons who commissioned works by artists like Da Vinci and Rubens. Since we credit Da Vinci for The Last Supper and Rubens for The Descent from the Cross, and not the patrons who commissioned these works, we ought to "credit" the deep learning algorithms for the "art" they produce rather than crediting the humans who "commission" this "art." >

… and we credit the camera for the photos it produces, not the photographer?