Hello.
I prefer 1c of the latest batch. I see no harm in using the word 'design', but I would look for a bolder, more extended typeface. This one is a little 'straight' in comparison to the initials and needs to fit the chunkiness of the descender of the 'g'.
I would also consider colouring the 'm' or the 'g' to make more of the overlap before you commit to a final logo.
I created an identity which has some similarity in terms of the construction of the initials...
ENC/ GTC/ GSC Utilities | Logo identity | Paul Cartwright Branding