Here's a bit of extra evidence that the KIIS poll was abit leading in its formulation (by reminding the reader of the "accumulation" of Russian troops)
The probability of a full-scale Russian military invasion of Ukraine is rated by 19% of Ukrainians as high, 33% as a medium, and 20% as low. The corresponding poll was conducted by the “Rating” sociological group on February 16-17 .
According to the survey, 19% of respondents rate the likelihood of a full-scale military invasion of Russia in Ukraine as high, 33% as a medium, 20% as low, while 25% believe there is no threat. In addition, 28% of Ukrainian residents are absolutely sure that Ukraine will be able to repel Russia’s attack, and 13% are not at all sure.
The poll concludes that in the last few days the assessment of the probability of a threat as high has decreased from 28 to 19%. At the same time, the confidence of citizens that Ukraine will be able to repel the attack if the Russian invasion takes place has increased from 58 to 64%.
The the KIIS poll was forcing a yes/no/dunno rather than more finely graded scale. Also, even back then, the results in the Rating poll(s) showed a fairly high degree of day-to-day variability, on the finer scale they used.
'Rating' also has this more details graph of the responses on that Q in previous polls
Depending how you want to parse that, their Feb 2022 results were not substantively different than those in 2018, for instance. OTOH those in April 2021 were different enough. A bit of googling shows that's when the first Russian buildup was reported. So, I guess one conclusion from that is that repeated/continuous media reports of troop concentrations at the border [over 2021-2022] had an effect in the public mind of reducing the threat perception, i.e. a kind of habituation response. OTOH, the Feb 16-17 poll was conducted right after the press/Russia announced they were pulling some troops back from the border. So maybe the change from the Feb 12-13 poll was just a reflection of those news.