The two terms are often used interchangeably, but LMD is a specific case of DMD. DMD is a group of additive manufacturing processes that adds material alongside the heat input simultaneously. The heat input can either be a laser, electron beam, or plasma arc. So LMD is the one using lasers in particular.
Direct Metal Deposition may include Chemical Vapor Deposition (CVD), Sputtering, electroplating, etc. Laser Metal Deposition involves laser as an energy source to accomplish deposition. Where the laser strikes, photochemical and/or photothermal reactions happen in the gas phase or the solid surface resulting in a metal being formed as a product and deposited on the substrate. Organometallic compounds in the gas phase are good precursors to this process.
Laser metal deposition (LMD) is a unique technique, combining laser and powder processing, which enhances material utilisation by enabling manufacture of high precision near net shape components from powders.
Laser direct metal deposition: theory and applications in manufacturing and maintenance https://www.sciencedirect.com/topics/engineering/laser-metal-deposition
Laser metal deposition is a generative manufacturing method for metals. Internationally, it is generally known as "laser metal deposition", abbreviated to LMD. People also talk about "direct metal deposition" (DMD) or "direct energy deposition" (DED). The process is easy to explain. The laser creates a weld pool on the component surface. Metal powder is automatically added via a nozzle. Beads that are welded together are formed, resulting in structures on existing base bodies or entire components. The process is used in industries such as the aviation and aerospace industry, energy technology, petrochemicals, the automotive industry, as well as medical technology. TRUMPF customers benefit from a wide range of lasers and laser systems, process expertise, and services for numerous applications. This means LMD technology can also be combined with laser welding or laser cutting.
As far as I know, both processes are the same and lie in the category of DED AM processes. For more details, please read: https://www.sculpteo.com/en/glossary/dmd-definition/