I know of no such distinction. #2 doesn't sound natural to me, although there may be times when it would make sense.
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SnappyDo native speakers strictly distinguish 1. from 2.?No. Not at all. For example, you might be saying that someone is left-handed by saying "He tosses a ball with his left hand". There's nothing about the determiner the (as opposed to his) which signifies handedness.
SnappyMy example was in the past tense. On the other hand, the above examples are in the present tense.Tense would not make a difference.
Can I understand that "He tosses the ball with the left hand." is acceptable if he is left-handed?