刻む: chopping an onion and engraving names
Distinguish the onion being finely chopped from people’s names to be engraved on a stone monument. Read the material, inscription, location, and planned timing.
The chopped material is an onion
「玉ねぎを細かく刻むと目に涙が浮かんだ。」 means “Chopping the onion finely brought tears to my eyes.” 玉ねぎを identifies the material being chopped, while 細かく describes the fine chopping. 刻む takes its chopping sense in this example.
Tears appeared in this particular chopping event. The sentence gives the material and that reaction without establishing everyone’s response or a cooking standard.
The planned inscription is people’s names
「石碑には村を救った人々の名を刻む予定だ。」 means “The names of those who saved the village will be carved on the monument.” The inscription consists of the names of people who saved the village, and its location is the stone monument. The Japanese 刻む予定だ states a plan to engrave, rather than completed engraving.
人々の名を names the inscription, while 石碑には identifies where it is to be engraved. The sentence concerns names on a monument rather than carved figures of people or a metaphor about engraving memories in the mind.
Use the object and location to identify the sense
The onion example has 玉ねぎ as the material to chop. The monument example has 人々の名 as the content to engrave. The same verb takes different senses with those objects and their locations.
The first example reports a reaction during chopping, while the second presents future engraving as a plan. Read the objects and the stated timing together rather than treating both actions as completed.
What is the object of 刻む in the stone-monument example, and is the engraving planned or completed?
Show the answer and why
The names of the people who saved the village are to be engraved on the monument. 刻む予定だ states a plan to engrave those names; it does not report the inscription as already finished.