Just skimming both directive again, (btw you can find the full originals ones for free on Eur-lex)
I found the bit which I think is the cause for the confusion.
Basically you must apply the content of the LVD (or use it as guidance), but its covered in the MD so basically you don’t need to announce the LVD.
“1.5. RISKS DUE TO OTHER HAZARDS
1.5.1. Electricity supply
Where machinery has an electricity supply, it must be designed, constructed and equipped in such a way that all hazards of an electrical nature are or can be prevented.
The safety objectives set out in Directive 73/23/EEC shall apply to machinery. However, the obligations
concerning conformity assessment and the placing on the market and/or putting into service of machinery with regard to electrical hazards are governed solely by this Directive.”
Just to help with your research I have linked the manuals from the two big players with their DOC inside.
Apply MD and LVD
The German version (pretty old but only one on their site)
Only mentions LVD
Apply MD and LVD
However this one only has the MD.
Non of them mention RoHS, but one would assume they are using RoHS components. (We have too)
Interestingly Trotec manual from 2019 no longer mentions the LVD and now revisiting the text i can see the interpretation that does indeed imply the LVD does not need to be listed. I will update our documents for the next batch of machines.
Oberland have been using the FabCore for a while now, seem quite happy with it i get the occasional photo or newspaper clipping from them 
How big a laser cutter are you looking for?
We have a FabCore Plus coming soon, Assembling the first this week 
Basically its the FabKit Mk6 but assembled and looks like a FabCore.
Looking for somthing even bigger, faster and more powerful then the FabPro, but he is quite a bit more expensive and will not be ready for another few months.
Curious what tips Teja gave 
and if GF respond with a DOC 
If you have any practical questions about the systems we employ for safety I’m more than happy to walk you through them.
Btw if you ever want to read the EN standards you can get them from Estonia for a very reasonable price https://www.evs.ee/et/
Kind regards,
Bonne