Ok, so my experience is limited to being a hobbyist doing field recordings of musicians 20+ years ago, and trying to expose my son to this sort of thing, and being a fan of a few youtubers. So take what I say with that grain of salt....
I think the short answer to "which microphone?" is the usual permaculture answer: "it depends", which can be an invitation to consider more context. And like in permaculture, the summary of that conversation is "it's more about the technique and skill of the practitioner than it is about the tools" and "there isn't going to be a once-and-done answer for each different shot in your video".
GOALS. I love your "fibre prep - washing wool" video. I'd seen it before and thought it was good, but watched it with different eyes and ears when you posted here what you were wanting it to be. I'm going to assume you want to shoot how-to videos for youtube that don't make the beginner mistakes, starting from where this video is. Better than that (netflix-level, hollywood-level) is really not achievable without putting in your 10,000 hours and having a crew.
Recorded video and audio is always giving the viewer a somewhat synthetic experience. When you are there, you can look around, you can tune out the chickens, etc. Just by the nature of recording, the viewer/listener cannot do those things. For example, the chickens probably didn't annoy you when you were there, but they are too loud when you sit down to edit. So the filmmaker does things to compensate. And our viewers have seen lots of film and video, and have expectations around how those things are done. In general, most video producers are going for audio that adds to the experience, without drawing undue attention to itself.
Also, everything is a compromise. "I don't want any wind sound at all." -- this is not literally achievable of course, but it takes some experience to know what will be "good enough". It looks like your video was not really in a super windy environment, but in-camera microphones are just notoriously bad for wind noise. Likely a dead-cat wind sock would be good enough for this situation, but maybe not on a more windy day--you'd have to think about shielding the mic (maybe with your body or a wall), and/or shooting a different time/place, and/or editing the worst of the wind bursts out, or maybe replacing the audio all together. Similarly, every microphone and every recorder (including the one in the camera) add some noise ("self noise", "noise floor"), even in a totally quiet environment--it cannot be zero because physics. The amount any given equipment choice adds isn't relevant for your sink and kettle scene, but if you were trying to capture the sounds of carding for that other video, or birdsongs, or the sound of a shuttle thrown across a loom, it would be very relevant.
EXAMPLES from your video. Again, I think this was a great effort. But re-watching after you asked, here's what I noticed that I would probably want to change if it were me:
0:06-0:11 - the kettle and stove maybe don't need to be this loud. Maybe bring down the volume in edit. The chickens are too loud in the background--how to solve? How far away are the chickens? How measurable-with-instruments-loud are they in real-live, relative to the tea kettle? Getting away from the camera's built-in mic will likely help a lot. A more directional mic *may* help, but up against that wall (sound reflector) or if the chickens are quite close, it may not. Maybe the best is to experiment while shooting, stopping to listen to what you are getting. Does your camera support a headphone output for this purpose? Listening only later isn't likely to make you happy. Bringing your external mic closer to the stove and kettle is surely one thing to try.
0:14 - the video cross fades to the new scene, but there is a sudden change in the background audio. Not too big in this case, but it can be distracting. Best to avoid this audio difference when recording in the first place, but if not, then cross-fade audio in edit to hide it.
0:16 - first loud wind-on-mic noise. Possibly a wind sock on an external mic would have taken care of this, but still, some can slip through. If there are only a few of these and they don't cover important audio (such as speech, shears), then you can plop down some "background" sound over the top to hide the burst. You should always get some specially-recorded background sound from each location, even indoors ("room tone"), but you can also just find another quiet spot to copy. You'll likely need to cross fade and manually adjust volumes to get a good enough hide. Fancy edit software may have tools to make this easier. It's like removing a zit from someone's face in photoshop.0:16-1:00 - I don't see any need for camera audio here at all. You could substitute specially-recorded background sound to keep chickens and wind down to your desired level. That audio could be recorded at a different place, further from the chicken pen, and even on a different day.1:16-ff - Sink, kettle, spoon sound good to me. I think you're working to hide chickens, though? If there were a voice-over, then you wouldn't be able to hide the chickens with the music.2:57 - Rooster. Maybe one rooster crow when the only other sound is the sink faucet *might* be tolerable?3:06 - Rooster's gotta go. I'd have shot more video, maybe repeated the action or the whole scene. You could try to remove the rooster by pasting other audio over, but not likely 100% effective. If the rooster keeps up, he needs to be further away. I assume the sink isn't moving, so what if the rooster was in a crate and closed up in the garage for 20 minutes?3:14 - Reshoot due to rooster. I'd be ready to harvest him by now.3:45-end - Substitute background sound, as the visible actions are quiet ones.
(flick carder video) - I don't it would be possible to get usable audio of the (quiet) carding process in an outdoor farm setting with any type or $$ of equipment, at any experience level. Shoot indoors with a close-in mic, or prepare to re-perform the carding indoors later while watching the video and trying to sync your motions -- like Foley in the movies. Maybe hard for carding, but maybe easier for spinning? It depends.
EQUIPMENT. There are some really good audio options nowadays in the prosumer space these days. However, there isn't likely to be one that will do everything. An on-camera shotgun-type mic is pretty popular with youtubers, but is best for presenters' voices or other stuff that doesn't need to sound too natural, or have too much stereo field. The "shotgun" refers to the long, thin geometry of the thing, which is designed to be very forward-directional. This may or may not make the chickens quieter--you still need technique that considers the space you are in, and to listen to what you are capturing in the field, before you finish shooting. A single shotgun mic, may be "good enough" for everything you want to do right now, or it may not. A second microphone option for some shots may make a big difference, depending on your goals. For microphone-on-camera, RODE is the most popular brand, with many choices.
Another level would be a wireless mic system feeding the camera. This is almost de rigueur
on youtube these days to get the presenter out of the background mush--a wireless lapel mic on the presenter's shirt. But wireless can be used for other effects as well. It allows you to get the microphone close, but just out of shot (or even inconspicuously in-shot!). If you're working alone, this could mean clipped to a presenter's clothes, on an additional tripod, or taped to a porch post or something. Usually the wireless system can be used with different microphones. RODE is popular in this space as well.
The next level would be having a separate, compact audio recorder. These can go on top of the camera, on a tripod, or anywhere really. They can record themselves, feed into the camera audio, or both. They always have a headphone output. The most popular brand is ZOOM. They have a wide range and most models have decent microphones on the front, plus inputs for more external mics. This is a huge upgrade, and the price for what you get is almost insanely low. The audio quality is much better than the in-camera recorder, and the additional flexibility is huge. The cost is that you have to sync up the separately-recorded audio to your camera video at edit time. For a few sink-and-kettle scenes, that's no big deal, but for many cuts with tight requirements, it can become a lot of labor, though there would be workflow techniques to reduce this, depending on what your editor software supports. I hear that modern video edit software can do this automatically, but I haven't tried it, so I don't know how effective it is. An external recorder would give you the most flexibility for the first $200 you spend, and that's what I suggest.
One pitfall: there are products that try to leverage your smartphone. I don't think the inherent compromises are worth it, except in very special cases.
So, in conclusion, I think getting some equipment of some kind that you could practice with is the best way forward. If you decide this isn't something you want to spend time practicing (a totally valid choice!), then I don't think rental equipment will help. You'd be better finding a volunteer to do it for you in exchange for them getting the experience.
Sorry this got long. Writing it will probably be the most fun I have today.
. I thought there's so much good videography how-to on youtube these days, that a quick search for "audio for b-roll video" would get someone explaining what I tried to write above. There were many promising search results, but unfortunately, the ones I previewed were either low quality, or focussed only on ways of getting the presenter's voice. I'll keep looking as I have a chance.
Good luck on this adventure, and keep us informed! I'm subscribed to your channel!