[Moto X Q&A] KidJoe:
Now that I have some time, I just want to dive into this a little more so we both understand this a little better... I'm no expert on this stuff, I'm just learning...

From what I've been able to find and read on the subject, Wifi Display, WiDi, and WiFi Direct are different. WiDi is an intel standard that has been out for a few years. Its used by intel based PC's and laptops to use a TV as a monitor without wires. Wifi Direct can connect two wifi devices directly, and can be used to transfer files. Wifi Display I'm still learning about as it appears to just be an underlying component to Miracast and Miracast Certification, not something used separately.

from the 5 Frequently Asked Questions on -> http://www.wi-fi.org/content/search-page?keys=Wifi display#.U2-IAIFdUhE

[B]What is the difference between Miracast and Wi-Fi Display?[/B]
Miracast is the brand for the certification program operated by Wi-Fi Alliance. Devices that pass this certification testing can be referred to as “Miracast devices”. Miracast certification is based on the Wi-Fi Alliance Wi-Fi Display Specification. This is the underlying technological specification developed by Wi-Fi Alliance members, and is copyrighted and owned by Wi-Fi Alliance.

[b]How is Miracast related to Wi-Fi Direct?[/b]
Wi-Fi Direct allows devices to connect directly to each other, without the need for a Wi-Fi AP, and requiring just the push of a button, the entry of a PIN, or tapping two NFC-capable devices together. Wi-Fi Direct allows source and display devices to discover one another and provides the underlying device-to-device connectivity for Miracast. Miracast builds upon Wi-Fi Direct with mechanisms to negotiate video capabilities, setup content protection (if needed), stream content, and maintain the video session.

Widi is NOT Miracast, see the above WiDi link I supplied. WiDi and Wifi Display are not listed on the certification documents for your TV and the X, only Wifi-Direct. In addition to Wifi-Direct, the certification sheet for the X also lists Miracast-Source.

So with the LG web site claiming Widi and Wifi Display, they might not have received certification for it. Why can they show the wifi alliance certificaiton on their box? Maybe because they received certifications for the other parts (like wifi-direct and connectivity)?

There are various levels of Wifi Alliance certification. Basic connection, Basic interoperability standards, backwards compatibility, etc. They ensure that you pick up a Wireless G card from belkin and it connects to your Netgear Wireless G router, or that your wireless N card can work with your B, G, or N router properly, etc. Just that "connect" needs Wifi Alliance certification, and that is enough to bare the symbol on the box. Then there are the "add-ons" like Wifi Direct, Miracast, etc.

My older LG BD390 and BD570 bluray players have the Wifi Certification logo on the box. Only wireless they can do is make a connection to a B, G, or N access point.

Think of it this way... You can buy a Buick Verano... get it equipped with or without Nav, with our without leather type interior, with or without sunroof... but any way you buy it, its a Buick Verano. Yeah, its not the best analogy, but I hope it, and the bluetooth link for the older phones I provided, illustrates different configuration possibilities while still being able to have the "certification" logo.

Oh, and there is similar dis-function in the DLNA certification process!! (even though my BD390 and BD570 received DLNA certification and have the logo on the sides of their boxes, they do not properly adhere to DLNA standards so some DLNA servers like Serviio and others don't properly work with them)

Each vendor is going to blame the other. That is how it goes. They build their devices to a "standard" and get it "certified" so they think it must be the other guy's fault.

As suggested earlier, you could buy a real Miracast dongle for your TV. It will probably be more cost effective than replacing the phone.

Or you could by a Samsung phone. I think they still support MHL on the S5. Or they support their AllCast Share (which would need a device for your TV to make work).

All that being said, the Wifi Alliance Certification certificate for my Sony KDL-5HX750 is very similar to yours, except your TV supports 5Ghz A band wifi, where mine does not. Both only list WPA/WPA2 Security, Wifi B, G, N, Wifi-Direct, WMM, and WPS.

The X (XT1060/XT1050/XT941L) lists those plus support for Wifi Direct and Miracast Source.

So in the interest of this discussion, and to see what is possible, I tried Wifi Direct via the following -> http://www.sony-asia.com/microsite/bravia_i-manuals/FY11/GA/eng/HX920_DS/nt_wfdirect.html and I can make the connection. If I follow -> https://www.youtube.com/watch?v=C0wnkZUVDKs I get to the 1:20 mark just fine, and the connection is established. The video then demonstrates "throwing" content from the device. The built in video player and photo gallery apps on the X do not have these options, unless I can find an add on app in the store?

I need to better understand the statement of "media remote" in the video since Sony does have a Media Remote app in the PlayStore, but that has been replaced by what looks like a TV remote only app. Installing the older Media Remote (OLD) app from the play store, In DEMO MODE I do get throw/catch options on my phone. I will have to get time to try and use this while connected to my TV. Its Mothers day, so that will have to wait.