This is the dechak, the oppression. And the prooftext is a citation of Shemot 3:9,
|ט וְעַתָּה, הִנֵּה צַעֲקַת בְּנֵי-יִשְׂרָאֵל בָּאָה אֵלָי; וְגַם-רָאִיתִי, אֶת-הַלַּחַץ, אֲשֶׁר מִצְרַיִם, לֹחֲצִים אֹתָם.||9 And now, behold, the cry of the children of Israel is come unto Me; moreover I have seen the oppression wherewith the Egyptians oppress them.|
What is the role of this prooftext? It clearly does not demonstrate that lachatz means dechak. After all, the prooftext does not use the word דחק, but just לחץ. Rather, the function of this prooftext is to show that לחץ appears in the full narrative as it appears in sefer Shemot. And it follows, in the text, after the demonstration of anyeinu and amaleinu.
How do Chazal know that it means דחק, oppression? Familiarity with Mishnaic Hebrew, as well as with Biblical Hebrew. Thus, for example, the following pasuk in parashat Balak:
|כה וַתֵּרֶא הָאָתוֹן אֶת-מַלְאַךְ ה, וַתִּלָּחֵץ אֶל-הַקִּיר, וַתִּלְחַץ אֶת-רֶגֶל בִּלְעָם, אֶל-הַקִּיר; וַיֹּסֶף, לְהַכֹּתָהּ.||25 And the ass saw the angel of the LORD, and she thrust herself unto the wall, and crushed Balaam's foot against the wall; and he smote her again.|
It clearly means pressing. So too the words are juxtaposed in a pasuk in sefer Shofetim (2:18):
|יח וְכִי-הֵקִים ה לָהֶם, שֹׁפְטִים, וְהָיָה ה עִם-הַשֹּׁפֵט, וְהוֹשִׁיעָם מִיַּד אֹיְבֵיהֶם כֹּל יְמֵי הַשּׁוֹפֵט: כִּי-יִנָּחֵם ה מִנַּאֲקָתָם, מִפְּנֵי לֹחֲצֵיהֶם וְדֹחֲקֵיהֶם.||18 And when the LORD raised them up judges, then the LORD was with the judge, and saved them out of the hand of their enemies all the days of the judge; for it repented the LORD because of their groaning by reason of them that oppressed them and crushed them.|
I am unsure whether this was affliction from taskmasters or whether it was a subjugated position in society. (Looking to Akkadian, we have the word chalatz, which is semantically related to לחץ, via metathesis. It means pressing, as in pressing grapes, and appears in proximity to the cognate for sechita, squeezing.)
Earlier, in Shemot 3:7, prior to pasuk 9, there is mention of other forms of affliction:
|ז וַיֹּאמֶר ה, רָאֹה רָאִיתִי אֶת-עֳנִי עַמִּי אֲשֶׁר בְּמִצְרָיִם; וְאֶת-צַעֲקָתָם שָׁמַעְתִּי מִפְּנֵי נֹגְשָׂיו, כִּי יָדַעְתִּי אֶת-מַכְאֹבָיו.||7 And the LORD said: 'I have surely seen the affliction of My people that are in Egypt, and have heard their cry by reason of their taskmasters; for I know their pains;|
So the different shades of suffering may be highlighted even local to sefer Shemot.
We might notice a shift here in topic, in terms of the nature of the suffering. If we do say that there is a semantic link between the previous two terms, anyeinu and amaleinu as prevention of conjugal relations and destruction of male children who would result, then lachatzeinu as simple oppression is unlike the others.
Looking to the Sifrei, we immediately notice that the derasha on לחצינו is absent. The Gra's emendation of the text places a derasha on lachatzeinu into here. But perhaps there was some derasha that was more on topic, or no derasha was made, but a derasha was brought in to fill the gap, so as to form a continuous running commentary on the pesukim. (But then, the whole derasha on vayar et anyeinu seems brought in from "agada" and placed in parentheses, and the derasha on vanitzak, and the prior verse, is wholly absent.)