Which Siri on which device does what?

MacWorld has a whole article about this in 2018. From the summary

The iPhone and iPad are the kings of Siri. Every request I made was answered best by these two devices, and the responses always matched up (though the language often differed).

The Apple Watch is the slowest to respond. Often times it would give me an I’ll tap you when I’m ready message before offering up an answer several seconds later, even when connected to Wi-Fi and tethered to an iPhone.

The Mac is the worst Siri device. While it's fine for general knowledge questions, It can't set timers or alarms, can't control smart devices through HomeKit, and can't add songs to a playlist. Plus there's no (official) Hey Siri functionality.

HomePod is by far the best at listening and responding, and it's clear that Apple wants this to eventually be the high-water mark for Siri.

Siri on Apple TV is woefully underrepresented. It's missing key Siri components, including Hey Siri summoning and audible responses. And its answering left much to be desired, even when asking about movies.

Siri really needs a personality boost. Apple has added some fun interjections when asking questions and its responses are very natural, but it doesn't play games, read stories, or tell jokes nearly as well as Amazon Alexa or Google Home.

I know if I ask Siri to set a timer, my phone will set the timer.

I'm assuming that you have "Hey Siri" enabled because I do this many times per day.

But, why not the iPad?

The iPad is fully capable of setting a timer via Siri. I just set one on my iPad by invoking Siri with the home button. It's also capable of doing so via "Hey Siri" as well. If yours isn't doing so - either via "Hey Siri" or manually, I suggest we look at this in another question, because that in and of itself is a technical issue.

If I ask Siri to open Google Chrome, which is on all three devices, which device responds?

I actually tested this. Siri opens on the device that responds first. Putting my iPhone next to my iPad and saying "Hey Siri" both devices responded however, it was always my iPhone that took control becuase it had the Siri "tone" first. When that tone sounded, Siri on my iPad closed. For the record, Handoff and Continuity was enabled.

For some reason, I don't have "Hey Siri" on my iMac with Catalina (10.15.5). I can only invoke it by clicking the Siri icon or pressing the hot key assigned. However, I would assume it functions in the same manner.

As for putting a priority on the device, decide which one you want to respond to "Hey Siri" enabled on one device and not on the others. It's easy to invoke. I have mine on my phone since I use it to control Apple Home (my lights, fans, etc.). I don't enabled Hey Siri everywhere to avoid confusing things.

There's no different version of Siri for different device types. Siri is Apple's digital assistance service and is made available over an active Internet connection. What this means is that a request made to Siri is processed on Apple's servers, and you cannot use Siri without an active Internet connection.

With that said, Siri does not generally differentiate or use a different service based on the device type the request is coming from. Siri may or may not allow you to perform an action based on if that feature is available on the said platform.

For e.g., you can set a timer on an iOS app as the feature is available into the built-in Clock app. But no such functionality is available by default in macOS.

Additionally, it is possible to integrate Siri functionality in 3rd party iOS apps but the same is not available for macOS apps.

So even if it's possible to invoke a function available in a 3rd party app using Siri on your iPhone or iPad, the same would not be possible in macOS apps from same developer if they have an app on both the platforms.