Context. White-light stellar flares are proxies for some of the most energetic types of flares, but their triggering mechanism is still poorly understood. As they are associated with strong X and ultraviolet emission, their study is particularly relevant to estimate the amount of high-energy irradiation onto the atmospheres of exoplanets, especially those in their stars’ habitable zone. Aims. We used the high-cadence, high-photometric capabilities of the CHEOPS and TESS space telescopes to study the detailed morphology of white-light flares occurring in a sample of 130 late-K and M stars, and compared our findings with results obtained at a lower cadence. Methods. We employed dedicated software for the reduction of 3 s cadence CHEOPS data, and adopted the 20 s cadence TESS data reduced by their official processing pipeline. We developed an algorithm to separate multi-peak flare profiles into their components, in order to contrast them to those of single-peak, classical flares. We also exploited this tool to estimate amplitudes and periodicities in a small sample of quasi-periodic pulsation (QPP) candidates. Results. Complex flares represent a significant percentage (≳30%) of the detected outburst events. Our findings suggest that high-impulse flares are more frequent than suspected from lower-cadence data, so that the most impactful flux levels that hit close-in exoplanets might be more time-limited than expected. We found significant differences in the duration distributions of single and complex flare components, but not in their peak luminosity. A statistical analysis of the flare parameter distributions provides marginal support for their description with a log-normal instead of a power-law function, leaving the door open to several flare formation scenarios. We tentatively confirmed previous results about QPPs in high-cadence photometry, report the possible detection of a pre-flare dip, and did not find hints of photometric variability due to an undetected flare background. Conclusions. The high-cadence study of stellar hosts might be crucial to evaluate the impact of their flares on close-in exoplanets, as their impulsive phase emission might otherwise be incorrectly estimated. Future telescopes such as PLATO and Ariel, thanks to their high-cadence capability, will help in this respect. As the details of flare profiles and of the shape of their parameter distributions are made more accessible by continuing to increase the instrument precision and time resolution, the models used to interpret them and their role in star-planet interactions might need to be updated constantly.