I discovered this on KH-Vids, which is running XenForo 1.2.4, but I also tested this on KH-Flare, which is running XenForo 1.3.0, and the issue happens there as well.
This was fixed for the forum list, but when a username is changed, it is not reflected on personal conversations. Below there are screenshots: some are from KH-Vids and some are from KH-Flare.
In these screenshots from KH-Vids, we'll use the Staff Team Page and Application Information PCs. The first one contains me and Farron as participants and the second one contains me and Mixt as participants. Farron is currently Llave on the forum, his default username, and I requested a change and was changed to Cortex yesterday from Toan, but as you can see, that was not reflected on the personal conversations listing and inbox alerts.
In these screenshots from KH-Flare, I just changed my name from Amaury to Cortex, but as you can see, that was also not reflected on the personal conversations listing and inbox alerts.
Doing some further extensive testing, it seems this only affects invited participants. As you can see in the KH-Flare screenshots above, in the conversations I initiated, my name changed from Amaury to Cortex, but in the conversations I didn't initiate (e.g., Test Conversation), my name did not change from Amaury to Cortex.
To test that and make sure my theory was correct, I changed the username Test Account to Test, and the username updated on the conversation I initiated as Test Account, where Test Account was the conversation starter, but not on the conversation I initiated as Cortex, where Test Account was the invited participant