There are a few possible answers and facts here that build on each other:
1) A previous comment of Rashi just before on 75a, already spelled out the case by saying we do not learn out "from the red dyed skins of rams"..etc. because they did not need to be specifically killed by "shechitah". (Apparently they were not needed to be eaten or offered, but merely used as furnishings.)
Therefore, Rashi did not need to use the full three word title for the rams by the time of his second comment. So, any "diyuk" is inconclusive.
2) The wording in Rashi says "Taking of a life (as a melachah) occurred in the Mishkan by the reddened rams, "Tachash" and "Chilazon fish"." The word "skins" would be a clumsy addition to the sentence, since skins don't have their lives taken, but rams do. Therefore, we can assume Rashi shortened the wording because he already spelled it out before, and its better shortened. It is fair to say the reader would still understand "reddened rams" as the short version of the Torah's "reddened ram's skins".
3) However, you ask that Rashi could have simply written "rams" ?! That would be even shorter and satisfy the above without any confusion at all?
So, there is an argument between Rashi and Rav Hai Gaon. Are the first eleven melachos for the purpose of growing/processing herbs for dyeing the tapestries and ram skins, or for making the show breads and other mincha offerings? Rashi holds it was for dyes and not for the bread. See Rashi, 73a "HaOpheh" and Gemara 74b. Rashi says that baking was not in the Mishkan.
The argument is about if an Av Melochah can only be something used to construct the mishkan (Rashi) or can even be used during an offering brought in the Mishkan (Rav Hai Gaon).
Therefore, if Rashi had simply said "rams" as an example of taking life, it could be misconstrued as any ram (like a "korban" as well as the red rams). That would be incorrect, since slaughtering an offering could never be counted as an Av melochah according to Rashi. Therefore the proper language for this comment was "reddened rams".
4) "Reddened rams" could also hint at something else in Rashi. The Gemara Yerushalmi in Perek Klal Gadol, says that the melochah of dyeing (colors) was accomplished with the skins of the "reddened rams". This means that before the animal was slaughtered, they beat the rams with sticks to cause bruising which would redden the skin. Rashi, (as you know in Chumash-see OP), does not bring this Yerushalmi, but rather says that the skins were dyed red after slaughter.
In fact, there is a three way argument about the act of causing bleeding on Shabbos. What melochah is it? a) Dyeing the skins b) Extracting the liquid called blood from the flesh which is Taking life "Ki hadam Hu Hanefesh"(parshat reeh) c) Weakening the living through extraction of blood
Tosfos on 107a (beginning of Perek Shemoneh Sheratzim) and here on 75a near the bottom (Ki Heychi), point out that taking even some blood is the melochah of taking a life, even if the creature does not die. (some say the Rambam only lists it as extracting a liquid, so he holds that taking partial blood is not taking a life, since nothing died.)
Rashi himself on 107a, brings a first opinion and a second. First, Rashi says that causing a bruise is taking life. Second, he brings an alternate opinion that the bruise is dyeing the skins.
Tosfos explain that Rashi really holds that the first opinion is the main one.
Now, it is proper to say that the opinion which Rashi brings in Chumash is true, along with the Yerushalmi's opinion, which is also true. The rams were bruised while alive, slaughtered, skinned, and then the skins were dyed further. However, Rashi comments in Chumash, to show that the (main?) melochah of dying is not like the Yerushalmi (bruising), probably because we usually pasken like the Bavli (regular dyeing of skins after death soaked in solution)instead.
Therefore, in Rashi's comment to 75b, he is teaching us an additional point. "Taking of life occurred in the Mishkan with the "reddened rams"...etc.
By saying that "reddened rams" are included in taking life, instead of just "rams", Rashi shows us that the slaughter of the ram, AND the "reddening" (read "bruising") were both part of the category of taking life (as opposed to the Rambam who holds its only extracting, and as opposed to the Yerushalmi who holds it is the example of dyeing).
(Note: The Bavli also includes causing bleeding with dyeing the skin. However, this is reflected in the opinion of Rav on 75a-b, as bleeding during/after "shechitah" that reddens the neck wound area. However, this coincides with death. The Yerushalmi counts bruising as dyeing which occurs while the animal is still alive as you know.)
Therefore, for all of the above, Rashi used "reddened rams" in Shabbos 75b and "skins dyed red" in his commentary on Chumash.
Hope this helps, :)