Digital exposure is a complex topic, surrounded by lots of confusion and clouded by obfuscation.
The sensor records light in a linear fashion. For each pixel, for each exposure, a number is reported from 0 to 16,384 (typically; there are other possibilities). Each pixel has either a red, green, or blue filter in front of it and so each pixel records (mostly) only red, green, or blue photons. (The filters relatively wide, mimicking the sensitivity of the nerves in our eyes. A very few blue photons will make it through the green filter and even fewer through the red filter, and vice-versa. And blue-green photons will make it through both the blue and green filters, but not as many as blue will make it through blue and green through green.)
Exactly how many photons it takes to result in a pixel recording, say, the number 1126, will vary from sensor to sensor, but exactly twice as photons will cause the sensor to record 2252.
Further, prior to that number getting recorded, different amounts of electronic amplification gets applied to the signal before recording. The number of photons that causes the camera to record 2252 at ISO 100 will cause it to record 4505 at ISO 200, and 9008 at ISO 400.
Just as turning up the volume knob on your stereo causes more and more distortion, so does the electronic amplification of the camera's ISO setting. What in a perfect world would result in recording 9008 at ISO 400 might result in 8996 in one pixel and 9012 in a pixel next to it, both of which received the exact same number of photons.
Now, let's assume you're in a perfectly-controlled studio environment with a flat subject and even illumination. And, you're photographing a scene that includes a light trap (a hollow black-lined box with a small hole at the top), an 18% gray card, and a piece of Teflon thread tape (which reflects 99.9% of the light that hits it).
Your picture, if perfectly exposed, would record 0 for the light trap, 2^14 * 0.18 = 2949 for the gray card, and 16384 for the tape.
Remember those filters? They really mess things up.
First, "white" light (which doesn't exist, but never mind) is a mixture of light of all frequencies, but not in equal proportions. Second, for very good technical reasons, your camera has as many green-sensitive pixels as it does red and blue pixels combined.
So, the perfect exposure now becomes one in which the brightest channel (which is virtually always the green channel) has those values indicated above and the other two channels fall where they do. Typically in daylight, the blue channel will be about 2/3 stop underexposed relative to green, and red will be just over a stop underexposed relative to green. Those ratios shift depending on the light source, and figuring out exactly what those ratios are is what white balance is all about.
So, let's say that we nailed the in-camera exposure and the gray card came out at 2949 in the green channel. The red channel might have come out at 1475, and the blue at 1946. Presumably, the light trap still came out at 0 for all three channels, and the thread tape would have come in at R=8192 G=16384 B=10912.
To get your perfectly white balanced perfect exposure, you'd double all the red values, leave the green values alone, and multiply all the blue values by 3/2 -- after which the red, green, and blue values for the three objects under discussion (all of which are different shades of gray) would be the same.
What about the ball bearing in the scene? Much of it is just mirroring the rest of the scene, but there's that specular highlight that's reflecting the light source, and that part is a lot brighter than the thread tape. In our photo, all the parts of the ball bearing that are exactly as bright as the thread tape get recorded as R=8192 G=16384 B=10912, but the parts brighter...well, green can't get any brighter, so it stays at 16384, but let's say that R=10912 and B=14549. The camera is telling us that it's a lot more magenta than it really is. And an even brighter spot (but not the brightest spot) has all three channels maxed out at 16384. We're back to white, but it's telling us that this is the same white as the thread tape, which it clearly isn't. And what about this other part of the scene where we've turned up the brightness on our lighting?
...and that brings us to how digital exposure actually works in the real world.
Specifically, all camera meters on the market actually tell you to underexpose the scene. My 5DIII, for example, underexposes by about 2/3 stop. The raw processing software (both in-camera and Lightroom, etc.) then applies an equivalent amount of digital overexposure to compensate. You could make a spreadsheet of all the values recorded by the sensor, multiply by (roughly) 3/2, then multiply red and blue by whatever you need for white balance, and the numbers would (basically) match the numbers coming out of the JPEG.
What that does is give you an extra 2/3 of a stop of headroom for your highlights. It also means that something more sophisticated than a simple linear multiplication can preserve more visible detail in those highlights.
There's no such thing as a free lunch, of course. The worst noise is always in the shadows, and applying 2/3 stop of digital boost will make those shadows that much noisier. All things considered, though, those shadows are awfully clean these days, so the tradeoff of having a bit more headroom for highlights is well worth a bit of invisible noise in shadows.
There are some additional caveats. Virtually all RAW processors (including the one in the camera that makes JPEGs) apply all sorts of other modifications to the data before you see the image. Most significant is the gamma adjustment; the sensor records in linear gamma, but the rest of your workflow is set up for gamma 2.2. Linear gamma images displayed without proper adjustment look very dark and contrasty. Next most significant is color profiling, which is yet another can of worms. And then there's an s-curve generally applied for contrast and "pop," there's almost always some sort of "secret sauce" (the picture style) to give a certain "look," and then there's all the knobs that you can fiddle with yourself (contrast, saturation, shadow boost, highlight recovery, and the rest). The end result is a very complex mathematical transform applied to the data.
And because that transform is so complex, that leads to the most unfortunate factor of all. Applying post-exposure digital exposure adjustments anywhere other than to the initial RAW recording of the data is going to amplify and otherwise interact with all those other adjustments, and generally result in chaotically unpredictable behavior. Modest changes generally don't have much of a visible effect, but the results with more dramatic changes can be...well, more dramatic.
I don't know if Camera Raw / Lightroom apply their exposure adjustments (with that top slider) before or after all the rest of the calculations. I would hope they're smart enough to do it before, but I've long since given up on ACR for color-critical work. I know for certain, for obvious reasons, that Photoshop itself is going to do anything like that after everything else has been done.
So, my recommendation for a general-purpose workflow is to target your exposures to the same as what the camera's meter is set to; that's almost guaranteed to be the best compromise between preserving highlights and reducing shadow noise. That's not to say that you should blindly trust the camera's meter, of course; meters can easily be fooled. Rather, get to know how your meter works with a perfectly-lit gray card and try to achieve that level of exposure. How to do that in practice is the usual challenge of the photographer...perhaps you'll use an external meter calibrated to your camera's meter, perhaps you'll use Adams-style spot metering, perhaps you'll gauge the histogram, perhaps you'll judge it from the preview image, whatever. But the point is to learn how to get the camera to expose the scene the same way its meter would under ideal metering circumstances.
If you can do that, you'll get the best compromise of preserving highlights and reducing shadow noise that your camera is likely capable of, and you won't have to deal as much with all the mathematical funkiness that goes on behind the scenes with digital development. Your images will be basically right straight out of the camera.
In scenes where you're still blowing highlights that you care about and have excessive noise in the shadows, you should first strive to fix the light by any and all traditional photographic means -- wait for the Golden Hour or add fill flash or use scrims or reflectors or whatever. If that won't work, either HDR of some form or a larger format camera is your ticket.
With a particularly convoluted workflow in controlled environments, you can more intelligently apply the "ETTR" concepts...but, even then, you're generally best off adjusting exposure as I described above so that the green channel is right on the 1.0 gamma graph. How to do that is much more involved than is reasonable for this forum...it involves analyzing ICC profiles built from a linear UNIWB development of the RAW image, and software (such as Raw Photo Processor) that lets you specify the channel multipliers yourself....
P.S. If you were to shoot, for example, a ColorChecker exposed exactly according to your camera's meter reading of a gray card, and it doesn't look almost identical to an idealized ColorChecker such as you can find at Bruce Lindbloom's site, then you've got a problem somewhere with your camera's meter. But if both your photo of the ColorChecker and Bruce's look too dark, then it's much more likely that your monitor's brightness is off, or that something other than the camera is to blame. b&