In my experience, the difference between various salts has little to do with flavor, once you've moved beyond iodized table salt and bulk kosher salt, and assuming we aren't talking about salts that are flavored by additions like herbs or smoke during processing.
So limiting the discussion to natural, high quality finishing salts, the differences are mainly texture and color. Some salts, like Maldon, are flaky, while others are large pyramids or cubes, and others tend to a small grain size and hold on to a bit of moisture. Each of these textures can bring something special to a finished dish. For example, flaky Maldon adds a delightful crunch, while another salt might adhere better to a French fry.
Color, like the pink salt you mention, is used pretty much for the visual interest. And there is nothing wrong with that. Simply save it for a dish where it will be noticeable. For example, those pink grains would look amazing on a chocolate truffle, or a savory meringue.
If anyone thinks they can actually taste the difference among unflavored finishing salts, I'd challenge them to do a triangle test with those salts dissolved in water (in equal amounts by weight) so that texture and color isn't confusing the issue.
Morton's Tender Quick is a fast cure salt meant to be mixed in with ground meats or used for curing thinner cuts (or fish). Hams must be brined for days in order for the salt solution to penetrate deeply through the flesh and prevent spoilage. It is important to follow good curing protocol and use the amount of salt as requested in the recipe.
Sodium nitrate, or pink salt #2, is not essential from a food safety perspective for whole muscle curing but is for dry-curing various salumi.
Best Answer
No, these are completely different products.
As you indicate, so called "pink salt" is a mixture of sodium chloride (regular salt) and sodium nitrate (or sodium nitrite) for curing meats, tinted pink to distinguish it from regular table salt. it allows relatively accurate small batch curing, as in home sausage making.
Himalayan pink salt is a naturally mined rock salt, whose impurities can tint it a variety of colors, including pinkish, from iron oxide.
Hawaiian pink salt is another naturally harvested salt whose red clay impurities tint it a pink color.