Logic quantizes notes relative to the start of the MIDI region. Most of the time a MIDI region starts at the beginning of a bar or beat, so the vast majority of quantizing would work as expected even i you didn't know about this condition. Even if a MIDI region started at the beginning of a sixteenth note (but not on a bar or beat), and all your quantizing was to 1/16 or less, then it would still work as expected. The workaround is to simply extend the beginning of the MIDI region to some more rounded value (like a bar or beat).
This condition is actually a feature that makes sense in various contexts: If you have a musical run in a MIDI region, but you move the MIDI region to some non-rounded position, then you would likely want any quantizing to be relative to the region, not to the global grid.
Of course, if you snipped a "regular starting" region in half at some non-rounded position, then the 2nd region would therefore start at this non-rounded value. And it would likely be your intention that any quantizing to this 2nd region be to the global grid - hence the problem you're encountering.
There is a handy keycommand called "Set Optimal Region Position Rounded By Bar" that will change the MIDI region boundaries so it starts/ends on Bar lines (it also makes regions shorter if there is no data to justify the long length). There is a similar keycommand called "Set Optimal Region Position Rounded By Denominator" which does the same thing at whatever user-defined denominator you had set (/16 is the default). This second keycommand would likely make most sense if A) you only quantize to 1/16 or less and B) if you've got MIDI regions jammed up beside each other such that the "round by bar" method would cause an overlap which might make things visually confusing.