The Kwee - van Woerden (KvW) method for the determination of eclipse minimum times has been a staple in eclipsing binary research for decades, due its simplicity and independence of external input parameters, which makes it also well suited to obtain timings of exoplanet transits. However, its estimates of the timing error have been known to be of low reliability. During the analysis of very precise photometry of CM Draconis eclipses from TESS space mission data, KvW's original equation for the timing error estimate produced numerical errors, which evidenced a fundamental problem in this equation. This contribution introduces an improved way to calculate the timing error with the KvW method. A code that implements this improved method, together with several further updates over the original method, is presented as well. An example application on CM Draconis light curves from TESS is given. Its eclipse minimum times are derived with the KvW's method's original 3 different light curve foldings, but also with 5 and 7 foldings. The use of 5 or more foldings produces minimum timings that are of substantially better precision. The improved method of error calculation delivers consistent timing errors which are in excellent agreement with error estimates obtained by other means. In the case of TESS data from CM Dra, minimum times with average precisions of 1.1 seconds are obtained. Reliable timing errors are also a valuable indicator for evaluating if a given scatter in an O-C diagram is caused by measurement errors or by a physical period variation.