Searching for a chord name

Hi guys...I am currently working on putting names to chords I have been using without knowing their names.

So..I have this dominant 7th chord starting on the 6th string...
But I remove the 2nd finger (so basically I am barring all stings..and adding the 5th of this chord).

In the example I added...its a D dominant 7 chord.
What would it be called if I remove the 2nd finger.

D #9th ? D dominant 10th? Or D dominant 3rd?

Thanks for the help on this crazy quest of mine.

Ids Schiere said:

The chord in the diagram is called D7 if you remove the second finger it's Dm7. Effectively a dominant 7th only is a called a dominant 7th if it is in combination with a Major third otherwise it's just minor seventh

Click to expand...

Holy crap thank you Ids!
Im just trying to figure it out...the diagram shows the dominant seventh..by flattening the third (removing finger 2 ) its a minor 7th...damn I love this!