Red Dwarf (the star, not the TV series)

Red dwarfs don't emit just red light. The star 'color' generally refers to where their output peaks. So red dwarfs are strongest in emission of the red end of the spectrum, but they also emit yellow, green, blue, violet, etc. The actual color would be kind of an orange cast. I don't know if you've ever been in an area that's downwind of a forest fire. We had that situation a couple of times this year. The light, especially in late afternoon, had a distinctly golden or orange shade to it, compared to normal. Something like that.

Red dwarfs belong to spectral category M. As you can see here and here, that means they have a surface temperature of around 3,000 degrees Kelvin. By contrast, our Sun has a surface temperature of around 6,000 degrees Kelvin. There are plenty of light bulbs these days that list their "color temperature" in degrees Kelvin, i.e. the temperature of a hot surface that would emit light of the same color. (Typically far hotter than the actual temperature of the bulb itself, of course.) So you can get a pretty good idea of the color that way.

There would definitely be an orange cast, but all in all I think it would be quite faint. The human visual system automatically adapts to light sources with different color casts, so that light within a fairly substantial color range is all perceived as pretty much white.

If you're talking about the "twilight zone" between the light and dark sides, then the light would become much redder just due to atmospheric effects, for the same reason the light of the setting sun here on Earth becomes much redder. The thicker the atmosphere, the greater the effect. So I think atmospheric conditions might wind up making more of a difference to perceived color than the intrinsic properties of the star itself.

Oh man, I've always thought that a tidally locked habitable planet would make for a fascinating setting for a story. I even started outlining one once, but never followed through on it...

Anyway, as for your actual question: Yes, the light from an M dwarf would be noticeably redder, though Kevin is correct that the human brain is pretty good at automatically correcting for colored light (though sometimes this backfires, which is why that striped dress picture went viral a year or so ago). Kevin's suggestion of checking out light bulb color temperatures is a good one. Looking directly at the M dwarf, it will still look pretty white to the eye, but it would give everything on the surface of the planet a sort of sunset look to it. In your twilight band, the redness of the light from the star would be exaggerated by the atmosphere, so you would have very red sunsets and sunrises. But it'll never look like purely red light like in a dark room. Stars emit light basically like "blackbodies", which means they emit a broad spectrum of light (that's why you see a full rainbow when the sun's light is dispersed). The color of a star is determined by the wavelength where the most energy is emitted. The hotter the star, the shorter that wavelength and the "bluer" the star will look, but that's not the only wavelength that the star emits.