The correct cable to connect early 2009 iMac to 2017 iMac in target disk mode

1. I want to be able to transfer selected files from my early 2009 iMac running MacOS 10.11.6 to my new 2017 iMac running MacOS 10.15.3. The old iMac ports include an ethernet (10/100/1000 Base-T) and a Firewire 800, whilst those on the new include a gigabit ethernet and a Thunderbolt 3 (USB-C).

I've tried connecting the old to the new in target disk mode using an ethernet cable. All I get is a black screen on the new with two large white symbols - a Thunderbolt and a USB - slowly moving around like a screen saver, but no sign of the target disk on the old desktop.

2. I've been told by Apple Support I can only connect in target disk mode if I upgrade both iMacs to 10.15.5. For a start, my old iMac won't update beyond its present MacOS 10.11.6, my reason for buying a newer one.

They've also assured me that the two floating images are normal, unlike Use target disk mode to move files to another computer – Apple Support, which advises that "when your Mac enters target disk mode, it displays a large Thunderbolt , USB or FireWire symbol onscreen", the operative words being "a" and "or"; ie, one or the other, not one of each . . .

Elsewhere - ie, in https://support.apple.com/en-us/HT207443 - I read "If the cable from your external device doesn't connect to the Thunderbolt 3 or USB-C port on your Mac or iPad Pro, you might need an adapter".

It goes on to suggest that in my case, "a third-party USB-C to Ethernet adapter, such as the Belkin USB-C to Gigabit Ethernet Adapter https://www.apple.com/shop/product/HJKF2ZM/A/belkin-usb-c-to-gigabit-ethernet-adapter" would do the trick.

But there's no mention of Firewire.

3. Is this correct?

Also, bearing in mind my need to be able to transfer files selectively, is it the only possible solution?