How do I explain my most recent job dismissal on job application?

You are using the old supervisor as a reference for this new job application. As such, you should expect that the old supervisor will be contacted (if your application gets that far through the process). And if your old supervisor is a stand-up/ethical person, they will answer inquiries honestly. You obviously can't control what is or is not said in that conversation if it happens. But you could lower the chance of the conversation happening by not using the old supervisor as a reference--which might be problematic for other reasons. As you said, this is tricky. So I think the best presentation for you is to make sure that everyone potentially involved in the decision (your application, HR, new supervisor, old supervisor) is hearing a consistent truth when they communicate, meaning nothing confusing or contradictory pops up in the review. If you go the super bare minimum route then others will define for you what happened (former employee X was competent but sketchy, blah blah). So I would probably give them a plausible explanation to hang their hats on: "I was on medical leave. I was 100% focused on getting myself healthy. That was my priority. And as a result I allowed my communication with HR to lapse which was the cause of my dismissal". Something like that. You want to control the narrative.

I would be inclined to say something totally bland (and as noted, not in full sentences) like "Extended medical leave, now fully resolved" and only add additional detail if directly required. More detail is not usually your friend on this. So - if you haven't already, you need to reach out ASAP to your former supervisor to make sure they're okay with you putting them down, and answer any questions (minus medical ones, obviously) about why you did what you did and why you're past it. This bears repeating. I've had a few people list me as a reference without asking, and even when I'm unreservedly supportive it's awkward because there's no way to not have the surprise call from the prospective employer reveal that I wasn't expecting to be asked. And when I'm not supportive, I'm not very inclined to hide that in a surprise call, whereas if asked ahead of time I'd either demur or figure out a way of saying something true but kind.

My old supervisor knew about the issues I was having that prompted me to take leave, but even though they were supportive, even though I'm including them as a work reference [other than the disappearing act, I was a good employee], even though they are a totally stand-up/ethical person, I can't help but wonder what they'll reveal even if inadvertently, and it worries me. Let's separate the two things - it's very unlikely that a stand-up employer will reveal the reason/source of why you went on leave, but there's nothing ethical or stand-up about a co-worker leaving out vital information like you dropping off the face of the earth in the process of referring you. Life's full of allegiances and they're not just to former employees. The process will likely be entirely retrievable in an HR file anyways - the reasons for your leave may not, but the process for them terminating you surely will be. As an employer, this isn't necessarily you dead-in-the-water - but a cover letter explanation or in-person answer isn't going to do it. I will have to see that, before you applied for this position (even the day before), you did a mea culpa with your supervisor, offered them a chance to understand why you did what you did, and asked them to be a reference - and they said yes voluntarily. Reading this, it's not clear whether or not you've communicated with them since you left - and being put down for a reference without being asked is not okay, especially in this instance where the reference may not be something I'd be comfortable giving without an explanation. So - if you haven't already, you need to reach out ASAP to your former supervisor to make sure they're okay with you putting them down, and answer any questions (minus medical ones, obviously) about why you did what you did and why you're past it.

Nthing the idea that you call up your old supervisor and ask them frankly about recommending you. Not only "would you be willing to serve as a reference and state that my work was acceptable?" but also "I'm considering applying for another job in the department, and I'd like to honestly know whether you'd be willing to recommend me for that job. I definitely learned some lessons about how important it is to keep up on required paperwork even in difficult circumstances, but I worry that HR won't be on my side. Basically I don't want to stir up trouble, and one of my deciding factors would be whether you would be supportive if Mr NewSupervisor asks about me." (note that you're implicitly admitting fault, and saying you won't do it again.)

I don't have specific advice for you, but I would inquire whether you have really thought out your parameters. For instance, you ask, "Do I state the bare minimum that they would likely find out from HR anyway," which is a different proposition from stating "the bare minimum." When I fill out standardized forms, often I don't even use complete sentences. My point is not to necessarily encourage you to state the bare minimum—I can't say whether that's appropriate for your circumstance—but only to suggest that if you are considering "bare minimum" as one possible parameter, then make sure you really are pushing that out to bare minimum. Know your range of options.

You know that your old supervisor will be contacted so you need to be honest. You need to say you were involuntarily dismissed and if you need to explain, I would say there was a miscommunication about a medical leave or an extended medical leave. Maybe you don't want to go into it, but given the circumstances of being fired, you can't really not mention the medical leave thing. Are you sure about using your old supervisor as a reference? Have you reached out to this person and specifically asked if they would be willing to be a positive reference for you as you look to re-enter the workforce? Because if they are called out of the blue, it's probably not going to go the way you want. And even if you were a great worker, if you just disappeared like that, I'm sure it was frustrating and didn't leave a great lasting impression, so this person may not even want to be a reference. Whether you use your old supervisor as a reference or not, given the closeness of these departments, it sounds like the new place will be calling your old supervisor anyway. You should really try to plant the seed of positive feedback from your supervisor by giving him/her a heads up and having some sort of positive interaction.