As you already have realized, color management (or color in general) is at the very least an elusive topic, and needs lots of explanations, knowing lots of different notions about color theory, maths and, perhaps, common sense. I’m not going to enter that discussion (endless discussion, maybe…).
I’ll try to make easy explanations, as short as possible, but even then, this post is going to be lengthy. Sorry for that. And I’m going to explain things from the RawTherapee point of view. I have no clue about how darktable or other software manages colors.
As others have already said, you should always work within a color management environment if you care about showing the «right» colors. Not using a color managed workflow or exporting without an embedded profile is prone to errors and weird colors (it may not happen, but yo can’t be sure at all).
If you understand color management as «a way to convert colors between devices so all of them show the same hues», then the way it works is pretty simple (for us, users; it can be a nightmare for developers, though).
In a raw developer the image has certain colors, which we are processing and modifying to taste, but we can’t see them. They are in the so called processing engine. To be able to properly interact with the app tools, we have to see the image, so a conversion is made, from the profile the image is in (the working profile), to the display profile (that’s why a properly calibrated display is of utmost importance). Again: you will never ever see what is «in» the processing engine, but you need to get an accurate conversion of that image, so you can see as accurately as possible the colors the engine is dealing with.
If you are happy with the results, you export your image with a convenient profile, by means of another conversion from working profile to output profile, so colors remain the same as they were in the processing engine (let’s put aside out of gamut colors for a while).
Then, somebody else downloads your image, and by means of another conversion from the output profile to a display profile (the display used by that somebody), the image colors remain the same.
If you don’t embed an output profile, your raw processing app WILL apply certain profile to the image (in RT is sRGB, by default), but will not say anything to anybody about it. So it is encoded in certain color space, but nobody knows it. It’s like giving somebody some coordinates to find a treasure, but nobody knows which map they belong to.
Let’s say after that an application loads that image, and as there’s no profile embedded, it will ASSIGN an sRGB profile… There’s an important distinction between converting and assigning profiles: «converting» means recalculating the RGB values of each pixel, and even if they change, the color will remain the same. It is just that the coordinates of such color are different in a different color space, but the color remains the same. On the contrary, «assigning» means preserving the pixel values (the coordinates) and using a predefined color space. If the image has been encoded on sRGB and the assigned profile is the same sRGB, then you’re lucky. If not, then colors will certainly change. How much will depend on how different is the color space used for conversion from the one used as assigned color space.
Following the treasure example: you have given coordinates pertaining to Poland, but the treasure hunter starts using a map from the USA. Most probably he won’t find the treasure at all…
So, from my point of view you must always embed a color profile. That way you will at least be relatively sure about people using proper color management will see your image more or less like you intended. The rest will or will not see it like you wish, but you have no control over that. If you don’t use an embedded color profile, you won’t have any control over anybody looking at your image.
Now let’s talk about soft-proofing and the image rendered on display.
There is an unavoidable fact: your display will never ever show colors outside its own gamut. This seems a silly sentence, doesn’t it?
Usually your working profile will be a large one. Much larger than any display will currently be able to show. Professional display or not. You can’t see all possible colors from a large gamut. And usually you won’t see all colors of an output image, unless you have a good wide gamut display capable of showing 100% sRGB, and you save your image with an sRGB profile.
Why is this important? Because with your Dell display you will never see 100% of the possible sRGB colors (according to specs, it covers 99%). And if you wish to save your image with, let’s say AdobeRGB, then for sure you won’t ever see all the possible colors. The display usually will never show the output profile, even when soft-proofing.
In RT the conversion while soft-proofing may be like this (depending on the combination of buttons and settings): working profile > output profile > display profile.
Why is it like that? Because the display can only show its own gamut.
About browsers and the dreaded Chrome/Chromium (at least until version 85): some time ago I opened a thread about the convoluted way that browser handles (handled?) color management. In short: if you wish to be sure about Chrome rendering about the same colors as Firefox and RT, you better use a profile with a standard sRGB TRC («gamma curve»).
If for some reason you still wish to save your images as sRGB without an embedded profile, you better use an industry standard profile: sRGB IEC 61966-2-1:1999. This is the one which almost everybody will default to. But as I said, to me this is not advisable.
The profiles from Elle Stone are better in my opinion, but they will be slightly different from the previous one. I wouldn’t expect much of a change, but anyway, there will be some differences.
Finally, the rendering intents are there to cope with those out of gamut colors I didn’t want to talk about at the beginning. And they are important because, at the very least they are needed so you can properly see your wider gamut colors (from the working profile) within the narrower gamut of your display.
I have left out many topics and details here, but to process an image, maybe there’s not much more you need to know about. Well, it will depend on how accurate you wish your colors to be rendered 
Hope it helps.