Same linker, heavier modifier: plain attributive ties a quality or owner to the noun (紅的書, 我的書); here a whole verb phrase, a clause's worth of action, sits in that slot (會說德文的老師).
的 ties a phrase forward onto a noun (跑得快 is not this); 得 follows a verb and reports how the action turned out (他跑得快). 的 modifies a noun, 得 modifies a verb.
noun before the verb phrase: 英文老師的會說德文 → 會說德文的英文老師
relative word added: 老師誰會說德文 → 會說德文的老師
linker dropped after a verb phrase: 會說德文老師 → 會說德文的老師
English hangs the clause after the noun with who/that/which (the teacher WHO speaks German); Chinese front-loads the whole verb phrase before 的, before the noun, with no relative pronoun.