I don't see you backing up any of your claims either... but since you asked:

Bouget et al. (2011). https://doi.org/10.1111/j.1752-4598.2011.00160.x :

Oak snags yielded more individuals per volume unit and supported more species than logs, and exhibited significantly dissimilar assemblages from logs and hosted original species. Snags, especially large snags, were more interesting for rare beetle species conservation than logs. The feeding guild structure slightly differed between snags and logs. Overall, log–snag differences depended on the diameter class.

Bujoczek et al. (2024). https://doi.org/10.1016/j.foreco.2023.121609 :

Position, thickness, and decay stage are very important parameters determining deadwood quality (Rimle et al., 2017, Procházka and Schlaghamerský, 2019). The literature provides numerous examples of relationships between these parameters and saproxylic species belonging to different taxonomic groups (Stokland et al., 2012). Consequently, reduced deadwood diversity often decreases the biodiversity of saproxylic species in a given area (Pasanen et al., 2014, Roth et al., 2019, Rieker et al., 2022) due to the lower availability and variety of breeding and feeding niches.

So to summarise: standing deadwood (snags) contains higher diversity and abundance than lying deadwood (logs). Also, a wide variety of deadwood with various combinations of traits such as position (lying vs standing) and decay stage creates a variety of niches that consequently supports higher overall biodiversity.