Merriam-Webster refers to explanatory information that appears within a definition at the level of individual senses of the defined word as "an italicized label or guide phrase." Although the Eleventh Collegiate Dictionary's treatment of the noun open doesn't include "(with the)" or any other guide phrases, its handling of the adjective open includes several:
8 b (2) of a tone : produced by an open string or on a wind instrument by the lip without the use of slides, valves or keys ... 11 a (3): of a compound : having components separated by a space in writing or printing ... 12 a of an organ pipe : not stopped at the top b of a string on a musical instrument : not stopped by the finger ... 16 of punctuation : characterized by sparing use esp. of the comma 19 of a universe : having insufficient mass to halt expansion gravitationally
All of the italicized bits of the above block quotation are what MW calls "guide phrases." I suspect that the terminology is intentionally broad (and rather vague) to comprehend the many types of such explanatory information that a dictionary may need to use from time to time. In any case, Wiktionary's "(with the)" notation would certainly fall under the category of "guide phrase" in MW's terminology.
At another level, of course, you are free to call such explanatory information whatever you like. Context specifier, which Jim suggests in a comment beneath the poster's question, is one possibility; usage limiter and phrasal prerequisite and scope definer are others.