Bluetooth vs HDMI, whose DAC is it, or what does Apple TV actually do?

I have a basic conceptual confusion as to what is going on when streaming music to my home theater receiver. Both my Apple TV and laptop are connected to the receiver through their own HDMI ports.

1. What is Apple TV sending to my receiver? I assume it is digital data that the receiver's DAC and subsequent Codac's decode and send out to the amplifiers.

2. But, is there a difference in the digital data sent from the Apple TV to the receiver when the controlling source is from my cell phone, iPad, laptop? That is when I use AirPlay to choose Roon or Tidal content through the Apple TV.

Does it make a difference if the receiver is getting data directly from the Apple TV or is using Airplay from my iPad to the Apple TV?

3. There seems to be better fidelity, higher quality sound, streaming Tidal when I use my laptop connected to the receiver through HDMI directly. It sounds different when I use Airplay through the Apple TV. I am unsure why this would be the case. Isn't data data?
 
Apple airplay uses ALAC format which downsamples everything to 16/44.1
There's an element of truth in that, but it doesn't represent the full situation. ALAC is a lossless codec. It does not determine the sample rate or bit depth of AirPlay streams.
The AirPlay specification allows up to 24bit 96kHz, but Apple and its affiliated vendors have chosen most AirPlay senders/receivers to operate at 16/44.1.

Tidal "HiFi quality" streams are 16bit/44.1kHz. For the sake of this discussion, let's ignore their "Master quality" streams for now.

The HDMI specification allows different sample rates, but 99% of HDMI transmit/receive hardware is designed to run only at 48kHz, and multiples thereof.
And indeed, all models of the AppleTV output only 48kHz.

Now that we're done with the theory, let's follow your signal chain:

i) Direct connection.
Your laptop (MacBook?) receives the Tidal stream and will play it at its native sample rate of 44.1kHz, but must perform sample rate conversion to 48kHz before its HDMI transmitter can handle the data.
Your HTR receives this HDMI signal, de-multiplexes it, and passes it (still at 48kHz) to its internal DAC.

ii) AirPlay streaming.
Your iOS device receives the Tidal stream and will play it at its native sample rate of 44.1kHz, then pass it to AirPlay which will continue to handle the audio data at 44.1kHz, compressed losslessly withe the ALAC codec.
Your Apple TV receives this 44.1kHz AirPlay stream, decompresses the ALAC compression, and must then perform sample rate conversion to 48kHz before its HDMI transmitter can handle the data.
Your HTR receives the HDMI signal, de-multiplexes it, and passes it (still at 48kHz) to its internal DAC.

So the streaming signal chain has an additional link in the form of AirPlay, but this should not affect the audio data in any way (unless the original audio data is higher than 44.1kHz.)
The significant difference is which device is performing the sample rate conversion from 44.1 to 48kHz. Early Apple TV's - up to and including 4th gen - had a reputation for doing poor quality sample rate conversion. Even if you have a later model Apple TV, I suspect that your laptop is doing a superior job of this conversion.

But can you avoid this sample rate conversion altogether? Yes! And you will most likely hear an improvement over all of the above methods.
- Direct s/pdif connection - most MacBooks have a headphone jack which doubles as an optical s/pdif output. Connect this to the optical s/pdif input of your HTR using a mini-TOSLINK to TOSLINK cable. Now configure your MacBook for 44.1kHz output, and your audio should sound better.
- or AirPlay streaming - ditch the Apple TV for music (keep it for movies) and use an AirPort Express instead - these are still available on eBay. The AirPort Express operates at 44.1kHz, ideal for most music sources. Connect it to your HTR via optical s/pdif.
 
If I overwhelmed you with details, here's a summary:
with both of your current connection methods the signal will arrive at the HTR in a technically similar state, including, at one stage, sample rate conversion from 44.1kHz to 48kHz.
I suspect the difference you hear is due to the difference in how this sample rate conversion is performed; in one case by the Apple TV, in the other case by your laptop.
If you're asking simply for technical clarification, that's as far as the discussion needs to go.

But if you want advice on how to optimise your system for sound quality - this is, after all, a hifi forum - I further suggest that you will likely achieve an improvement by avoiding the HDMI interface altogether, instead connecting with optical s/pdif.
 
Ha! You're a good egg and I appreciate your willingness to unpack this. There is learning curve to be sure. My previous MacBook had Toslink but my new MacBook Air does not. I have been eying some stand alone USB>Toslink/SPDIF units but want to get a better grasp on what I need. I just started a Roon subscription and think that building up a Raspberry serve/streamer would likely make that converter redundant.

What is really interesting are all the Modes that one can choose after the Apple TV signal is received by the pre-amp processor. There is no reference in my Emotiva UMC200 manual on what they actual do other than to say that only the Modes that can be used show up to be used. They sound quite different and I can tell that not all speakers are being used in each mode or how the mode is shaping the sound. The Neo:6 C6 appears to do the best job with the Apple TV as the source. The others are:

MODE: Direct
MODE: STEREO
MODE: ALL STEREO
MODE: PLII MUSIC
MODE: PII MOVIE
NEO:6 M6
NEO:6 C6

When I Google these modes all I can find is that they are decoding standards. I have been unable to find more specifically how they are different and what speakers are being use..I would be much obliged to have another blast of tech. The last really helped pull the strands together into a coherent and understandable picture.
 
I just Googled "Emotiva UMC-200 Apple TV" and got this -
UMC-200 and Apple TV (4th gen) | The Emotiva Lounge
The UMC-200 selects the appropriate format based on what it receives. However it does allow the listener to choose some preferences within that format. You can use the Mode+ or Mode- buttons to flick through the available options for each input format.
I think the UMC-200 is telling you that it's being fed a stereo audio source, and wants to know what you want to do with the other 5 speakers!
"STEREO" seems obvious - output to the front L and R speakers only.
"Direct" ???
"ALL STEREO" should be the 2 stereo signals in parallel output to the centre L/R and rear L/R.
"PLII MUSIC" / "PLII MOVIE" should be used if your stereo signal contains the old Dolby Pro Logic II (DPL II) surround encoding ... but if no such encoding exists, it will probably "synthesize" surround channels - using either the "Music" or "Movie" profile.
"NEO:6 M6" / "NEO:6 C6" will refer to DTS Neo:6 surround processing -
DTS (sound system) - Wikipedia
similar to Pro Logic, DTS Neo:6 will synthesize surround channels from a stereo signal, using either the "M" Music profile, or "C" Cinema profile.
 
My previous MacBook had Toslink but my new MacBook Air does not. I have been eying some stand alone USB>Toslink/SPDIF units ...
Yes, a USB to s/pdif adapter would be a good idea to interface your MacBook Air to the Emotiva - but I see that such devices cost around $80, and I think that money would be better put towards an external DAC. Over on audiosciencereview they consider the best entry level USB DAC's to be the Schiit Modi 3 and Topping D30, both US$130.
Not to put down the UMC-200 - it's a well regarded device, and I'm sure its internal DAC is half-decent, but there's only so much DAC/DAC-output-stage technology you can pack into a multi-purpose unit costing only $600. The Schiit or Topping DAC is quite likely to beat it for SQ. Your connection would be:
MacBook Air - USB Type A to Type B cable - DAC USB input - dual RCA to dual RCA cable - UMC-200 analogue audio input
 
I literally created an account today just to thank all of you on this thread... for posting the question as well as the responses. I have googled this so many different times (100s) and only found junk like "how to connect your Apple TV". Thank you SO much for explaining the signal chain and hand offs. I will literally sleep better.

I'm on the hunt for a streaming device for Apple Music but I don't want to use a laptop OR AppleTV. I'd kill for an iPod that's made to sit in my audio rack like a CD player, that just has a little screen and does nothing but stream lossless into my DAC via Toslink/SPDIF. If you know of something like that please hit me up!