There is a lot of misinformation and a lot of confusion here but I think it probably comes mostly from not explaining things correctly. Or at least, not explaining it in a way that someone with very little knowledge to start with can understand. So, here is my take on it trying to make it clearer.
If you trust the meter, it makes no difference which mode you shoot in. Yes, you may get different results depending on the mode chosen, but ultimately, the camera makes the choice for you.
If you learn a bit more and start not trusting your meter, it still may not make a whole lot of difference. If you are in manual and you adjust what the meter tells you, it is no different from using ev compensation in one of the auto modes except that in manual you choose whether to adjust the speed or the aperture while in an auto mode, the camera chooses.
Now, metering systems have gotten much better over the years and today you can get mostly decent photos while spending you life in auto. If you want more control, you need to learn the basics of exposure but for the most part, it still will make very little difference which mode you shoot in as adjustments can be made in any mode.
It does however help to understand the different things involved in correct exposure when you photograph things that are not run-of-the-mill. For example, if you were to go and shoot a motorcycle race, it probably would be a good idea to shoot in a mode that lets you pick your shutter speed...
Hope that helps.
Btw, I only shoot in manual. But it has nothing to do with it being better. When I started in the business, there wasn't anything else and I'm so used to it that I work faster this way.
Edit = There is only one manual mode, manual. Whether some people want to call some of the other modes semi-auto or auto, or whatever, they are not manual.