Listen to voice messages on other devices than your console???

I would like see the ability to send and receive voice messages return for the X1; it's so much easier than typing. If you don't want to listen to voice messages from strangers (like squeaky voiced kids accusing you of cheating) then just don't listen to
them.

No Geek, its considered the same. Voice messages and voice mail. When you listen to voice messages, The Xbox 360's Dashboard (UI/OS) brings up a small box titled Voice Mail and you listen to the voice messages. All Xbox 1 consoles have the same audio codec.
Xbox 360 has VARIETY in ways you can send messages across Xbox Live. Xbox 1 only has ONE way to send voice messages across Xbox Live, and it's only via TXT. I'd consider going to the official Xbox feedback site. I voice message with MANY peps and friends across
Xbox Live. IT BRINGS MORE LIFE TO A MESSAGE than just TXT.

According to your photos, you set up the settings in the Xbox correctly.

Under "Display & Sound" you set that party chat goes to headset and speakers, which is correct, and SHOULD in theory provide chat voice to be audible.

And since you say your voice message was recorded indicated by the bars of the recording, your CTIA headset adapter is also the correct one.

It MIGHT be, that pre-listening to a voice message will send the voice of your message to the headset irregardless of the party chat setting over speakers and headset. But I have to check that on my console later this week. Currently, private matters keep
me more busy than I'd like.

The only thing I can currently say is, that your setup is correct and your settings are fine.

You SHOULD be able to hear game sound and OTHERS through your headphones, and others should be able to hear you speaking.

In order to test if this is an account, network or hardware related problem, do the following:

Open the messages tab in the Guide and compose a new message.

Instead of typing, press the Microphone icon and talk into your headset. This should create a voice message that you can listen to afterwards before you send it off.

You don't have to send the message, but are you able to listen to your own voice of that voice message?

If you are not able to listen to it, check if the headset works on other controllers, or if different headsets (don't) work on the same controller.

Also make sure that in the settings tab, that you have your headset and game volume set properly.

Then while in party chat, check that you didn't accidentally muted the chat with the X button on the controller.

I will recommend trying this directly in the console, because sometimes the features of the console are not the same as the Xbox.com features; for example, listen to voice messages.