Two iMacs using one external drive for Time Machine

Yea' changing names screwed up the old Time Machine backups, plus it cause the new Mac to start new backups.

Sounds like you need to completely Erase the Time Machine backup drive and start over.

1) If the external drive is 2 or 3 times larger than both Mac drives combined, then you could start new backups for both Mac's on the same drive.

2) If the external drive is not 2 or 3 times larger than than both Mac drives combined, then you should get a second external drive and use one for each Mac.

ref: Back up your Mac with Time Machine - Apple Support

Personally I recommend using two external drives.

The old external drive to backup the old Mac and a new external drive to backup the new Mac.

Assuming the external drive is big enough and you are willing to delete the old TM backups, the best way to back up multiple Macs to a single external drive is to (re)format the drive. GUID partition, APFS format and create two volumes, one for each Mac. They don’t have to be the same size. Then select the appropriate disk on each Mac as a Time Machine destination.

For example, we have 5 Macs in the house and I use a 4 TB SSD with 5 volumes on it to back them up (actually, I have a pair of them and update them every two weeks to swap one off site weekly; those are secondary backups while a local 10 TB NAS is the primary TM backup for all the Macs).

Here are the instructions to add a seperate volume to your Hard Drive for each Mac to use their own volume. I would recommend to erase the drive at this point and start fresh with new backups for each device on their own volume. The recommendation to have each volume 2-3 times the size of each Mac still applies for the backup.

Add, delete, or erase APFS volumes in Disk Utility on Mac - Apple Support

Besides the new Mac's interference, that might not work because you changed the name of the old Mac.

One way to find out, is to connect the backup drive up to the old Mac, then go to the Time Machine Icon on the Menu bar and select Backup Now.

While I suspect that it will backup. I'm concerned that it will be forced start new backups, which will consume even more space on the external drive.

You're welcome. No, not if the new Mac assumed, modified or adulterated the old backup.