Apple TV will only set at SDR (not HDR), TV shows HDR 10. I'm I really seeing HDR 10?

Yamaha Adventage RX850

That should support HDR (HDR10), if it has had the firmware update that brought that support.

My Apple TV will only go up to the SDR setting. It will not go to HDR. When I watch a TV show through my Apple TV, the TV says it is showing HDR 10. My question is this: is my TV really showing HDR 10 or just upscaled SDR?

Please also check your Chroma setting on Apple TV. Chroma "4:4:4" requires twice the HDMI bandwidth of "4:2:0" with zero benefit, at the expense of other video properties (such as HDR): The source streaming video will be encoded with chroma 4:2:0 anyway, so no need to set your devices higher.

Please also check your HDMI cables. Are they all rated for 18 Gbps or 48 Gbps? If not, then legacy cables could be a bottleneck for video properties.

Please also check the Match Content setting on Apple TV, as that could potentially override the video format setting.

Please also check the HDMI port settings on the TV and receiver. It may require an adjustment for high bandwidth use.

Try a Reset Video Settings, for Apple’s recommended settings for the currently detected devices.

When I bypass the receiver, the Vizio shows HDR and Dolby Vision.

Apple TV →[HDMI]→ TV →[HDMI (e)ARC)]→ receiver is a perfectly viable option, to get the best video output, especially for slightly older receivers that might limit HDR10/HDR10+/Dolby Vision or lack HDMI 2.1.

Is my TV really showing HDR 10 or just upscaled SDR?

If Apple TV is really outputting SDR (not certain as mentioned above [Match Content]), then the TV will have to show SDR within its HDR screen presentation, possibly doing a conversion (“upscaling”) of sorts.