This is a good discussion point.
The Hawaiian D7 is called the Hawaiian D7 because it is used a lot in Hawaiian music. No doubt most Hawaiians do use the other D7 as well, when required. If you are teaching people how to play Hawaiian music on a Hawaiian instrument, it makes sense to teach the chord shape favoured by Hawaiians, even if it is easier to play. So the voice of the chord is a very important aspect to consider when choosing to use it.
If you are going to teach people about the D7 chord you need to know that the D major scale goes D E F# G A B C# D, it doesn't go D E Gb G A B Db D as shown by a computer search which can't work out enharmonic notes.
The notes in the 2020 shape are A C F# A, if you are really keen on getting a D note you could try 2025 A C F# d. This gives you the low C voice as well as all the notes in the chord. You can exercise your little finger by tapping the D note at fret 5 on the A string and see how it sounds.
Two other close by D7 chords are 2223 and 5655. You can learn all these chord shapes and you will find uses for them all.
The 2020 shape is usually only called D7 when it is used in the context of a D7 chord. If you play it in a progression which does not have a D7 context, it might have a different name.