LW displays notifications about replies and private messages in the same place, mixed together, looking the same. Note that top-level comments on your articles are also considered replies to you (this is a default behavior, you could turn it off, but it makes sense so you will probably leave it turned on).

This has the disadvantage that when you post an article which receives about 20 comments and someone sends you a private message, it is very easy to miss the message. Because in your inbox you just see 21 entries that look almost the same.

Suggestion: The easiest fix would probably be to change the appearance of private messages in your inbox. Make the difference obvious, so you can't miss it. For example, add a big icon above each private message.