While the MIDI editor is open, displayed MIDI event positions can be edited or quantized to fractions of a MIDI tick (ppq). But the actual saved, loaded, and played-back positions are rounded to the nearest MIDI tick.
In your video, the temporary display of a perfectly fractionally quantized MIDI note is just how it is displayed. The actual note start event may be up to 0.5 ticks on either side of the grid line. (This is almost certainly not a perceivable difference; 0.5 MIDI ticks at 960 ppq and 120 bpm = about 1/4 of one millisecond.)
The only reason you are seeing this at all is because your MIDI recording does not start on the beat. After recording, you trim the left hand edge of the recorded MIDI to the grid, but that does not affect the actual start time of the recording. Instead, you could either start the recording on the beat and then trim it to a later beat, or else start the recording wherever you want, then trim the left edge to the beat and glue the media item. Either of those actions will cause the underlying MIDI recording to start on the precise beat, which will then mean that all later beats will fall exactly on an integer number of MIDI ticks, and the quantized display will exactly match the playback.
Really, though, it doesn't make a practical difference.