I have a beginner level camera, Canon T3i. I understand a higher Digic processor helps improve higher ISO with lower noise, and faster continuous shooting. Would upgrading to a higher Digic processor help produce more authentic color?
A faster processor can contribute to things like faster AF computations, better tracking algorithms, automatic digital lens corrections, faster burst rate, better SOOC jpg noise reduction, improved video processing and quality, etc.
But it can't really pull any extra information out of the sensor that wouldn't be read normally. Also, you can't upgrade just the processor in your camera, you would need to just get a newer camera at that point.
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For your question about noise, your computer has a much more powerful processor than any camera (and it is not limited by needing to instantly process things in order to be ready for the next shot). If you are not happy with the jpg results from the camera, just use raw and develop the photos on your computer.
For example, DPP is free from Canon and will let you fine-tune noise reduction settings. There are also other raw developers as well. DXO is payed (or pirated) and has one of the most advanced noise reduction algorithms out there but it can take seconds or minutes to process depending upon your settings and computer specs. Obviously that does way more than any camera processor, even the newest ones.
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A major thing about "authentic color" is going to be getting your white balance correct. If it is off, the whole image may be tinted or biased towards a different shade. Your camera will have several white balance modes (auto, shade, tungsten, Kelvin scale, etc.) to choose from as appropriate. If you are outdoors, generally auto or one of the weather presets will get you pretty close. If you are indoors and know the kelvin rating of your light source, you can just set your camera to that.
If you have time to set it up, you can also do the most accurate option which is to set a custom white balance. This is where you photograph a grey card (or a white wall) under the same lighting as your subject and then calibrate the camera to that light temperature or color cast specifically. This needs to be done separately for different light sources, so if your light changes or you walk somewhere else, you would need to redo the process.
If you manage to mess up somehow but still have the raw files, you can set your white balance in post by either choosing a weather preset, kelvin value, or the eye-dropper tool to pick a pure white object (or neutral grey) object in your scene. You can also adjust color cast with a jpg, but it doesn't work as well and is a bit more work. Typically it is better to get your white balance right on-site though so that you don't have to guess later.
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The other part of "authentic color" has to do with the sensor itself and the camera's firmware. There are several color/tone presets that you can set depending upon your goals. These are described here:
https://global.canon/en/imaging/picturestyle/style/index.html
Generally speaking, "faithful" will try to capture colors and tones as accurately as possible while the other modes will punch stuff up biased towards various subjects (ex: portrait, landscape, etc.). Be aware that "faithful" may have less saturated colors or lower contrast than "standard". Most cameras will also let you fine-tune or adjust these picture styles in the menu settings (ex: contrast, saturation, etc.).
Bear in mind that there is nothing unique about each Canon preset that you couldn't achieve on your own playing with curves and color balance, it's just a preset to save you time. If you wanted, you could also use Canon's picture style editor to create your own preset to apply either while editing raw files on your computer or possibly add directly to your camera (not sure if the T3i supports this).
https://global.canon/en/imaging/picturestyle/editor/index.html
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One final thing to note (and probably less important or relevant to you) is that different lenses can have warmer or cooler color casts or various levels of contrast. So if you have a a preset all configured how you like for one lens and then you switch to a different lens, it might not look exactly the same or might require different editing depending upon taste.
