Bardic Inspiration and DM awarded inspiration are completely separate things
First, lets take a look at RAW in this case. Bardic inspiration reads as follows
You can inspire others through stirring words or music. To do so, you use a bonus action on your turn to choose one creature other than yourself within 60 feet of you who can hear you. That creature gains one Bardic Inspiration die, a d6.
So we can see from this ability that a character inspired in this way is not technically inspired, but rather gains an inspiration die. As such, it is not counted as being "inspired".
So, RAW supports this, but what about how the rules are intended to function? Well, it's the same thing here too. DM awarded inspiration is meant to be a reward for players who role play their characters very well, while Bardic Inspiration is meant to be a class feature that let's a bard's music fill their allies with confidence. As such, it seems that the reason for the wording of the two abilities was done that way with the intention of keeping the two abilities seperate