You are correct in your interpretation. Let's break it down: NEC 500.8(B)(1) emphasizes specificity

The key is that the equipment must be identified not only for the Class (e.g., Class I or Class II) and Division (e.g., Division 1 or Division 2) but also for the specific substance present.

Class I vs. Class II

Class I environments involve flammable gases or vapors, while Class II environments involve combustible dusts. A piece of equipment designed to prevent explosions in a Class I environment does not necessarily provide adequate protection against ignition of combustible dusts.

"C1D2 only" means exactly that

Equipment marked only for "Class I, Division 2" is only suitable for Class I hazards. It has not been evaluated and labeled to protect against the specific hazards associated with Class II environments (combustible dust).

The lack of C2D2 markings is critical

If the equipment doesn't have markings for Class II, Division 2 (or appropriate Groups like Group E, F, or G), it's not certified for use where combustible dusts are present. The manufacturer has not demonstrated that it's safe in those conditions.

Therefore, your conclusion is correct

A DC power supply rated solely for Class I, Division 2 (C1D2) is not compatible with a Class II, Division 2 (C2D2) environment. Just because it's explosion-proof for gases doesn't mean it's dust-ignition-proof or otherwise suitable for dust. It's a common misconception that a Class I rating automatically covers Class II. It does not.

In summary, the equipment needs to be specifically rated and marked for the Class and Division that matches the hazard present. If it's only C1D2, then it is only rated for vapor and gas environments and not suitable in a dust environment. Flag for review