DateTimeOriginal is not defined by the EXIF standard as specifically the shutter-open or shutter-close moment; it only says it is the date/time when the original image data was generated. In practice, this appears to be camera-dependent.

From the reported tests here, both a Canon EOS 70D and a Nikon D850 recorded DateTimeOriginal at the start of the exposure, including for a long bulb exposure. But that does not guarantee the same behavior for all cameras or firmware versions.

Also, EXIF time is usually only recorded to the nearest second, and overall accuracy depends on how well the camera clock is set and whether it has drifted.

For precise timing, the best approach is to test your specific camera: synchronize its clock, make a long exposure while observing exact start and end times, and compare them to the EXIF timestamp. If you need higher precision than a camera clock can provide, use an external trigger or timing source with a more accurate clock.