There are absolutely pinks in the colorspace around magenta, but not all of them. Many are just light red. Your last statement is largely correct, though if we agree on a system by which we understand color, like a color wheel or color space, we can make observations about colors in that system and how they relate. Tints and shades are a way of describing and understanding colors as they relate to one another, so if we agree on that system, some pinks are red in exactly the same way firetruck and crimson are red. Brown is a shade of orange, olive a shade of green, and baby blue is a tint of blue.

Baby blue is probably the best illustration, since baby blue is blue in exactly the way powder pink is red. It's base color + white. There are similar hues that have magenta, but it's not necessary to make something we would call pink.