Solution: Deictic

Answer: WORDLESSWEBBERS

This puzzle is about deciphering an alien spider language through a series of stories.
This puzzle can be thought of in two parts: (1) deciphering the language and (2) extracting from the webs of letters.

Deciphering the Language

Searching the title brings up a big hint: the concept of deixis. According to Wikipedia: “In linguistics, deixis is the use of words or phrases to refer to a particular time (e.g. then), place (e.g. here), or person (e.g. you) relative to the context of the utterance.” In fact, deictic words are the only words in the spiders' language. Every sentence consists of exactly one time, place, and person, each conveyed by a specific spider appendage.
Though it may not be initially clear, the spider with "I AM HERE" translates to, well, "I AM HERE". This Rosetta Stone is helpful for constraining the possibilities of what this language could be.
There are many ways to decipher the language. What is presented here is just one way to solve this step of the puzzle.

Step 1

Observe that only the blue legs and ball of silk change in the story. One of the gestures has the same blue legs as the "I AM HERE" spider. From the title of the puzzle, we can guess that the words will be deictic in nature. From this we can tentatively guess that:

here	there/nearby	away/off-screen

These guesses might change over time as we translate more of the language.

Step 2

In the first four panels, only the pink legs and apple change in the story. The orange palps are the same as in the previous strip and we are reasonably confident that the blue legs mean here. We can guess:

present	past

Based on Panel 5, Panel 8, our current knowledge, and the "I AM HERE" spider, we can also guess:

above/below	it	I

Step 3

See Page 5 of 8: With our current knowledge and the panels, we can guess:

further past	future	further future

Observe that the angle of the pink legs indicate how far in the past or future it is referring to.

Step 4

See Page 7 of 8: We can translate the story with our current knowledge. In addition to a new orange gesture, this story clarifies the definitions of some gestures we already know.

you	where you are	where I'm pointing

Step 5

Extracting from the Webs of Letters

First, translate the panels of the final story. On the web version, the story appears below the horizontal bar. On the printable version, it is on Page 8 of 8.
Observe that there are five webs of letters and there are five temporal tenses in this story: further past, past, present, future, and further future.
Also observe that there are exactly three statements for each of the temporal tenses: one referring to the right-facing spider, one referring to the left-facing spider, and one referring to the weird-looking thing.
What we need to do is to find the locations for all three persons at all five times. These map one-to-one with the letters on the webs. Based on how the story strips are structured, we can assume that the top will be further past, the bottom will be further future, with the rest in between.
Placing the letters on the underlines and reading top-to-bottom, we get the answer to this puzzle: WORDLESS WEBBERS.

Author’s Notes

• When the theme was announced along with its alien spiders, I knew I wanted to make a puzzle where you decode something, well, alien. Specifically, I wanted the puzzle to be culturally agnostic and use as few puzzlehunt conventions as possible. An English-speaking, puzzle-solving student in Waterloo, Canada should be on an even playing field as my non-English-speaking, non-puzzle-solving grandparents in Southeast Asia.
• The idea of the puzzle came from two sources. One was a video of a jumping spider mating dance (which you should 100% watch). The second was the title constraint: it must include the letters "D", "I", and "T". I thought that the spider mating dance looks like sign language and came across the word "deictic" while searching for a good title. "Perhaps this was how the Spinners communicated?"
• The first version of the puzzle was nearly identical to the version of the puzzle you see now. The main difference: it did not have the "I AM HERE" spider (it was only available as a hint). This was the feedback: "I can confidently say that I have a PhD in english and have taken university-level linguistics classes and it was still a challenge... with the hint!" Hopefully you had a much easier time!