Bernie and Lorien have hit most of your major point, Alea, but I'd like to weigh in from an energetic adjustment point. Let's call them assists, though, as the word we use for this is very important...
The first energetic assist I give, I give to the class as a whole as part of the pose instructions: breath guidance. It is constantly amazing to me to see people struggling to release in a forward bend drop inches at once when they move the focus from 'reach the floor' to 'breath in such and such a way' - I have two main ones I use, based off my own findings in self-practice.
The second energetic assist is individually given and just as Bernie says, is usually to check in with someone looking less than easeful - maybe looking around at other students, shifting a limb over and over in an attempt to find comfort or sensation, or doing some variation I've not guided them into or towards and want to make sure whether they have a reason that attends to their safety and the purpose of the pose in the sequence I'm guiding, or I've biffed on my instructions. This adjustment is reassuring them that 1)I'm not practicing, I'm attending to them in their practice and 2)supply them with my belief in their autonomy and innate body-based wisdom. I might offer props, reiterate or introduce them to one of the breath practices or move towards the third kind...
The third energetic assist looks more energetic: I tell students to place their hands on their body and send their breath there. This might be the belly in a supine posture, or an area where the sensation is really intense and, while they can still breath easily, they can see the benefit of yin in training us to be with safe discomfort and challenge without reacting and running away. I also explain how psychologically our body welcomes our own touch, rather than just bricks and blankets, and it increases our self-trust and self-connection, making poses come easier since the body and mind establish a relationship based on cooperation and support rather than dictation and expectation.
My fourth energetic adjustment I do all through class and may or may not initiate specifically with students. As a Reiki practitioner I found that my classes flow easier and with more joy and less effort if I take a breath before going in the room (usually while washing my hands, something I love to do right before teaching) and consciously open myself to the 'universal life-force energy'. I started really using this with my dedicated students when we were practicing new and more complex postures that were really assisted with gentle hands-on assists.
Before running Reiki, my hands on work was adjustments - trying to fix something from the outside looking only at the exterior of my student's form. When I began running Reiki, because it emphasizes self-responsibility and self-healing (the practitioner only supplies more energy, the recipient then uses as much or as little where it will best serve them, all this being on a 'higher self' level and pretty sub and un - conscious) I found that I really couldn't adjust. I was moving from a teacher as power holder to teacher as power sharer, in approaching students I was there to serve their highest good, not their ego (get deeper, do more in the pose) or my ego (get them further, teach them the fanciest variations). Also, I wasn't focusing on the body of flesh and bone, I was looking at how the flesh and bone body reflected and was in alignment, or not, with the energy body. Simple hands on the body, requests to breath into where the hands were, suddenly made HUGE differences to the poses.
I've had numerous experiences where I did a common hands-on assist - pressing on the low back during Balasana/Childs Pose - and had students come up asking what I'd done today because they felt all this energy. I smile as they go on and on about the epiphany and when they pause I say that I did nothing different, I was simply running energy as I always do and the difference today was that they were in a space to consciously receive and be aware of it.
For myself - it made teaching much, much easier. Less work for me due to more support from the Universe. And a way to subtly create a container for the class to be nurtured in.
Honestly, I was blessed to fall into my Reiki Mastery program and I feel very strongly that Reiki or a similar technique in energy education should be in the toolbox of every Yoga instructor. But even if you just consider what you are serving when you are drawn to assist students, you'll be able to alter and enrich the experience by aiming to do so only when it will serve their best interests.
Namaste and thanks for giving me a space to share that! Hope it helps in some way!