Two MacBook backups on one Time Machine - how to delete one Mac's backup?

I assume you are asking about Time Capsule since Time Machine is the software component.

The only way to do that is to erase the TM backup disk with AirPort Utility. Follow these instructions: Erase an AirPort Time Capsule in AirPort Utility on Mac - Apple Support

Erase the backup disk according to Erase and reformat a storage device in Disk Utility on Mac - Apple Support.

Attempting to erase files using the Finder either won't work, or will corrupt the existing backup anyway. In any event you don't want an unreliable backup. If you want another solution, then set that backup disk aside and use another disk for the new Mac's backups.

At some point will also need to remove the disk from Time Machine's Settings, otherwise TM will be looking for a backup disk that is no longer available. It will eventually complain about that so you might as well do it now.

Thanks for that additional explanation. Once a backup disk has been in use for a while, very little space is likely to be available to add a completely new source — your new Mac in this case. The reason is that TM will create as many backups as possible while remaining within the backup disk's capacity. When it reaches that capacity, TM starts to delete the oldest backups to make room for newer ones... perpetually operating in a state of very little free space. That's normal.

What you need to do is essentially the same: erase the backup disk. Or, at your option, completely erase the new Mac (since it's new anyway), and upon setting it up again, choose to "inherit" the older one's backups. That way, you will be retaining the older one's backup history to the greatest extent possible, while adding new backups as needed... the same way it's been working.

Which option you decide upon is up to you, but for what it's worth I always choose to inherit older backups. It's seamless to do it that way.

In my opinion the ideal solution is to purchase an additional backup disk. If and when you decide to retire the older Mac, you can use both backup disk drives for the newer Mac. Time Machine will back up to as many hard disk drives as you wish to provide. One and only one backup drive doesn't comprise a robust backup strategy anyway, because any disk can fail at any time, including backup disks. You never want to be left with no backups at all, even for a short time.

One question I neglected to ask: How did you populate the new Mac's content to begin with? Your choices are to migrate its content directly from the other Mac, or from its Time Machine backup. Assuming that backup is current, it makes literally no difference which of those two options you choose.

The other way is to set up the new Mac as new, without migrating its content from anything.

Yes in all likelihood that will corrupt those backups, rendering them useless, leaving you with none, not even those from the older Mac... a bad situation.

But I should have clicked "inherit" instead of "create new backup". This inadvertently led to my storage problem because if I had clicked inherit, then the TM would have simply deleted my oldest backup instead of preserving it.

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