When you go to a hypnotherapist or go up on stage with a 'tist, the context pretty freaking clearly says that you're in for some hypnosis. They'll probably even explicitly ask for your permission and then explicitly state when it's about to happen. They'll use words like "hypnosis","relax","sleep", and "deeper". And they'll do weird things like stick your hand to the chair. All the while, the meta level fact that you're "doing hypnosis" is very salient and it's a key player. That's overt hypnosis.
But there is another kind of hypnosis where you don't have to do any of that stuff and you can still stick their hand to the chair. You still use hypnotic techniques, but it all just looks and feels like a "regular", if a bit exceptional, conversation. That's conversational hypnosis.
The main distinction between the two techniques is the amount of attention you're drawing to the meta level of "this is hypnosis!" - either explicitly by saying "this is hypnosis!" or implicitly by pulling out your pocket watch and spinning spirals. Since the general idea of hypnosis is to put the labels down, this is generally not desirable. If you say "let's do hypnosis", then everything following the "induction" is tagged with "hypnosis". This changes how people respond. If you skip that and just go ahead and do the same thing, then it often isn't labeled hypnosis and you'll probably get different results.
The "hypnosis" label has some good to it, like increasing expectancy for amazing things and the ability to dissociate conscious from subconscious. However, it also brings up not-necessarily-accurate to wildly inaccurate conceptions of what is going to happen. These conceptions can give rise to many objections that don't really apply to the actual process - like "oh no! I'm going to get stuck in trance!" Or "you can control my mind!".
The "hypnosis" label doesn't just affect the way the subject responds to the suggestions, it also affects their interpretation of the experience after the fact. Consider the difference between "oh, the hypnotist had a hypnotic power that he used to get me to do that" and "oh, yeah he convinced me to do that instead and it made sense on a gut level". Since almost no one (hypnotists included) has any idea what "hypnosis" is, it's generally more helpful and honest to not call it hypnosis. Unless you really are putting them into a zombie like state and giving amnesia for the whole experience after having them quack like a duck.
To get into it a little further, it's not quite binary. That girl whose arm I stuck in a bar would not have said she was hypnotized. I never asked to do hypnosis - I just showed her something and she went with it, and it didn't fit her preconceptions of hypnosis. Yet it was obviously something. Not just a regular conversation, but not a deep conversation either. Perhaps if asked she would have called it a "mind game"? Either way, it wasn't labeled and the lack of labels didn't really affect anything there.
I aim for it to seem like a just a (perhaps very moving) conversation in retrospect. That way, when I help them make a change, it's more likely to be an ecological and congruent change. If it seems like mind tricks to them, then they are more likely to give up on making sense of it and the change might not stick. Again, it is the difference between "he tricked me into doing that" and "he convinced me that I should be doing that".
Because of this stuff, I've been moving towards using more "conversational" looking things even in my formal hypnosis - even with things that I internally categorize as overt "hypnosis". I'm using less "I'm going to hypnotize you now, okay?" and more "I'm gonna show you something cool, okay?". Less of the weird stuff like "sleep deeper deeper" and more of just describing the specific pieces and leaving the rest aside - "you know how sometimes you are just so interested that you just listen and absorb ideas in their entirety before you try to figure out if the story is fictional or if it really matches reality? Yeah, do that". In these respects, it's approaching conversational - but not getting toned down. I'm not getting soft and indirect. It's just more precise. The lack of "weirdness"-flag-raising-potential is a side effect of communicating so well that it makes sense even to their conscious mind. And no, that's not a downside. You don't have to "bypass" the "critical factor", you just have to make sure it does not object to what you're doing.
It's not that I'm trying to sneakily embed "hypnotic techniques" into conversation, it's just that there's no reason to say "sleep deep relax" when you can be more precise and get more consistent results. It's just that it naturally comes out sorta like a conversation. Sometimes it's a conversation with a novel experience embedded in it, but the point is that it's not an extra step to turn it conversational.
However, there are times when finding the driving force behind something and amplifying it results in something that doesn't quite look normal - and you'd lose something by trying to force it into a normal looking conversation. So in some respects, formal hypnosis is more free - it allows you to break more rules of conversation. You can be more repetitive with suggestions, and you can sometimes be more firm with them. You can use more distraction. You can leverage arm sticks and "hypnosis" to build expectancy to change. You can give amnesia for the problem ever existing - or give a false memory of amnesia for the problem having been fixed (did that one recently and it was fun :)). And some of this stuff is valuable.
You can get similar effects with conversational techniques. In that last case in particular, using acknowledgement instead of trance would have been slow as hell. I could probably have gotten it with normal in-retrospect-seeming-conversational-trance, but it would have taken effort and I might have struggled. It was just easier to conversationally suggest mind bending hypnosis had occurred.
It seems like "be conversational even in retrospect" is a good heuristic for congruent and ecological fixes - start with that. However, if you've tried that (in real or in simulation) and there's an identifiably bad reason that it's not working - then go ahead and get weird.