First, regarding the color terms--it's entirely possible that speakers of biblical Hebrew would've sorted the color of tekhelet into some other color, and the word simply referred to the dye. Much like how we might use the term "indigo" as a color, but it primarily refers to the dye itself or its specific color, and is a subcolor of blue.

A bit of background--the bible and subsequent Jewish texts reference a dye called "tekhelet". A number of garments worn in the Temple were dyed with this (and a sister dye called argaman, but more on that later). However, the bible also says that ritual fringes, tzitzit, are to be dyed this color (the exact way to do that is a whole religious question), though they can simply be white if the dye is unavailable (even when it was extant, it was not affordable for most people).

What's clear about the dye is that it was made from some type of sea-snail. The process was lost in Jewish sources sometime in maybe the 5th century, but its high cost and the movement of Jews away from the areas where this snail lived cause the collapse of the dying knowledge in Jewish circles. There are a bunch of similar dyes known to various Mediterranean peoples. It is presumably the same as either Royal Blue or Royal Purple.

First, the color. As you noted languages code colors in different ways, and the categories don't necessarily work nicely together. Sources don't really talk about the color, but they do sometimes compare it to other things. The Talmud says it is similar in color to the sea and the sky (which points towards blue, but it could be that they thought of the blue-purple spectrum as one color, and they were simply saying that the color is in the same range). The strongest evidence towards blue is that the Talmud reports that there was a dye which was virtually indistinguishable from tekhelet, called "kala ilan", which was a plant-based dye. This has consistently been identified as being indigo. Indigo is, of course, somewhere between blue and violet, but fabric dyed with it is most readily identified as blue (that's what makes blue jeans blue). This points strongly towards it being a bluish shade, not a purple or reddish one. It's important to note that the non-blue contender is "Royal Purple", which is a purple-red sort of color. The Septuagint refers to it as Hyacinth, which is a blueish-purplish flower. Those references are both relatively close to the time when the dye was still being produced--later Medieval sources have a vast range of descriptions of the color, from black (Maimonides) to yellow-green (Rashi), though these could be confusion about color term identification. While it's not proof, it's fairly strong evidence. So while exactly this would've fallen in the blue-purple spectrum is unclear, it is probably a particular sort of the blue-purple ancient dye, not the red-purple dye (no source calls it reddish).

Also, there's another expensive dye sometimes mentioned alongside tekhelet that is fairly clearly identifiable as a red-purple derived from snails, called "argaman". The LXX translates it as πορφύρα, which refers to the red-purple dye known to various ancient groups. Because they're mentioned alongside each other it would sort of make sense that they're similar, but there's no explicit Jewish source that says "these are the two dyes for things, the color is different but both are from snails, but not the same snails".

So, it would seem likely this dye is identical to Royal Blue, and was a blue-purple sort of color. Besides the massive similarities in description of the snail, it is unlikely that there would be two different snail-based dyes in the same color spectrum at the same time in the same place without any knowledge passing back and forth. Archeology here is enormously helpful--numerous dying facilities have been found, which contain a large number of shells of the snail used to make the dye. These show that the snails used for both these dyes were snails of the Murex or Hexaplex genus.

Anyway, now for a little more history. The dye was basically unknown in Jewish sources until the late 1800s when a Chassidic Rebbe decided to try and identify it. He identified it as being cuttlefish ink. While it is a mollusk that lives in the Mediterranean, there were issues with his identification. Cuttlefish aren't exceedingly hard to find, and there are no archeological or literary records of cuttlefish-based dye in the region (though the Razdiner Rebbe couldn't've known that). Moreover, the actual process used to make blue dye (and he was convinced the dye must be blue) was the process used to make Prussian Blue, where the color doesn't actually come from the cuttlefish ink--it just uses it as a source of carbon. The process was unknown to ancient people, and would've worked just as well with any organic substance. While this dye was not accepted as the authentic ancient dye by very many people, there are still Jewish groups that use it. Many use it not knowing that there are multiple potential dyes, and get it because it's cheaper. Another argument against it is that it is not colorfast at all, but descriptions of tekhelet make clear that the dye was extremely colorfast.

Enter Rabbi Doctor Yitzhak HaLevi Herzog. He was perfectly positioned to identify the dye, being both a Rabbi (and Chief Rabbi of Ireland) and having a doctorate in Chemistry. He wrote his PhD thesis on this subject. First, he identified the issues with the cuttlefish, and rejected it. He evaluated sources in other Mediterranean cultures, and early archeological findings of dying facilities. He suggested several possible dyes, but never replicated an exact dying process.

After that, the concept of figuring out the dye got revived after centuries of dormancy. People started messing around with snails and seeing what they'd get, and archeological evidence continued building towards a Murex/Hexaplex source of the dye. Early attempts could only get a lighter purple dye, not the deep blue-purple hue (I think--I've never seen a real description of the rejected dye besides "not blue"). But, someone tried doing the dying outside, and found that the sun actually affects the color, and you get get bluer shades that way.

Since then people have started actually making ritual items dyed with this dye (starting more than 20 years ago, but it was still uncommon 20 years ago). Given all the above it's likely the snail is right, but it's not really clear whether the color is. The color they make is like a blue jeans blue, which matches the description of it being like indigo, but is less purpley than other descriptions. For the reasons discussed at the beginning, exactly what shade it was is probably not an answerable question.