Lots of great info here already, and not much that I would disagree with. Here are a few comments based on my own experience:
High quality true macro (1:1 magnification or higher) is generally done with very small subjects that sit still for a relatively long time (flowers, sleeping insects, insect eggs or larvae, etc.) and is best with equipment such as a DSLR or mirrorless camera with a dedicated macro lens and flash.
Getting “good enough” photos for iNat purposes of subjects that may be a bit larger and move quickly and/or are easily disturbed is best with a DSLR or mirrorless camera with a telephoto zoom lens. Here you are trading off the ability to get high magnification for the ability to work quickly from farther away.
For example, a 60mm macro lens or even a 100mm macro lens requires your to be fairly close to the subject, so much so that bees, butterflies, and dragonflies will often be scared away before you can get close enough to get a decent photo. But when you can get close, they will take better photos, especially with flash.
On the other hand, a telephoto zoom lens (in the general 70-300 range) will allow you to get decent magnification from farther away, but for small things that sit still, the minimum focus distance of your lens will keep you from getting as close as you would like, so your magnification will be limited. (If going this route, check the minimum focus distance of any lens you’re thinking of buying. If it’s close to a meter or less, that’s good. If it’s two meters, look for a different lens.)
Two practical, inexpensive ways to go with the latter option (telephoto zoom lens) and also get close to your subject when needed is to use either a “close-up” lens that screws onto the end of your lens, or extension tubes which attach between your telephoto lens and the camera.
A close-up lens is simple to use, but its limitation is that it has a fixed distance you need to be from your subject, while many real-world subjects are best photographed from various distances.
Extension tubes take a lot of practice to get good with, and you end up getting more dust inside your camera because you’re constantly taking your lens off in the field to attach the extension tubes. But once you figure them out, they provide a lot of freedom to work from a variety of distances, to get a variety of magnification levels, and with a zoom lens they give a lot more flexibility for composing your photos (choosing how much background to include in your photos, etc.) If you’re going this route, be sure to get extension tubes that have electrical contacts (they are often called “auto extension tubes”). I recommend getting a set of three different-sized tubes that you can use separately or stack together. There’s a lot of info online about extension tubes and how to use them.
Some commenters mentioned focus stacking. If you’re not familiar with the term, I recommend reading up on it. Know that it requires taking many photos of a subject without the subject or camera moving. So it’s best for subjects that sit very still, and it’s often best done with the camera on a tripod, so it’s not good for efficiently taking photos in most situations out in the field with active insects.
You may find that autofocus is frustrating because it rarely focuses where you want it to (accurate focus is critical with macro photography because there’s rarely as much depth of field as you’d like). I almost always manually focus my daylight macro shots.
Similarly, autoexposre often fails because your small subject is lit differently from the background, and the camera takes into account the background lighting. So learn how to manually adjust your camera’s exposure compensation, and also how to select the area on the screen that your camera uses for autofocus and autoexposure.
If you don’t already know how to manually adjust your camera’s shutter speed, aperture, and ISO to achieve a desired balance of sharp focus, depth of field, motion-stopping, and noise, Irecommend reading up on it and then practice, practice, practice. Start out practicing on subjects that don’t matter, then move on to subjects that do matter, and be patient because it can take a while.
I find I almost never have enough natural light to achieve the aforementioned balance that I want. That can be addressed by used a flash, but my experience has been that flash photos often look “fake” and many options for flash equipment are bulky and may scare off my subjects because of their size, the bright white diffuser, the shadow they cast on the subject, etc. So I take almost all of my daytime photos with natural light.
One camera parameter to consider, especially if you will not be using flash most of the time, is the camera’s “low light performance” or “high ISO performance.” In other words, how much additional “camera noise” is produced as the ISO setting is turned up. You may read that modern cameras are so good that camera noise is no longer an issue. That absolutely does not apply to macro photography in natural light. I recommend reading reviews and trying to find a camera that is rated well for its low light performance or high-ISO performance.
One huge advantage of going with a DSLR or mirrorless camera that allows for changing lenses is that it gives you the flexibility to add or change to different equipment bit-by-bit, as you learn what works for you and as your budget allows. Maybe you can’t afford a flash right now, but in 6 months you can. And maybe you start out with so-so quality equipment and as you use it, you get a better sense of which components you would like to upgrade. For me, the most important component upgrades I’ve made have been higher quality lenses, longer focal-length lenses, and lower camera noise capability.
For your budget-conscious situation, I recommend getting used equipment from a trustworthy source – either a local camera store (if you have one), a big-name online reseller, or if you know someone who often upgrades their equipment, they may be willing to sell you their old equipment. If using an online reseller, be aware that they sometimes sell equipment that is not up to par but they’ll take it back at no charge to you. So I recommend fully testing it and if you know a photographer, asking them to check it out for you as well, before the return period expires.
You are seeing a lot of different advice here because there are multiple options that work reasonably well in different situations, but also because personal preferences vary. So if you can “try it before you buy it,” definitely do that. Perhaps there is a high school or community college with a photography instructor, or a camera store (not a big box electronics store), or a camera club near you. Folks there may be happy to show you what the various options look like, and they may even let you try them out for yourself.