Untangling old MacBook hard drive from new MacBook Pro

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Your explanation is somewhat confusing. You have two computers - an old MacBook and a new MacBook Pro. If you do not plan to keep the old computer, then you should do a factory reset as shown below:

How To Do A Factory Reset

Selection A should be used on computers that came with Lion or later when factory new. These models had no disks included when new. Selection B is for Macs that came originally with Snow Leopard or earlier. These models shipped with Software Restore disks when new.

A. Factory reset of your Mac - Apple Support

B. Factory Reset Your Pre-Lion Mac Follow these instructions until you get to Step 6 of Factory reset of your Mac - Apple Support. At Step 5 you will need a Snow Leopard DVD or the installer disc that came with the computer.

1. Boot the computer using the Snow Leopard Installer Disc or the Disc 1 that came with your computer.  Insert the disc into the optical drive and restart the computer. After the chime press and hold down the  "C" key.  Release the key when you see a small spinning gear appear below the dark gray Apple logo.
2. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Utilities' menu. After Disk Utility loads select the hard drive entry from the left side list (out-dented entry - mfgr.'s ID and drive size.)  Click on the Partition tab in the Disk Utility main window.  Set the number of partitions to one (1) from the Partitions drop down menu, click on Options button and select GUID, click on OK, then set the format type to MacOS Extended (Journaled), then click on the Apply button.
3. When the formatting has finished quit Disk Utility.  Proceed with the OS X installation and follow the directions included with the installer.
4. If you are planning to sell or give your computer away, then do the following: After you reformat your hard drive and reinstall OS X, the computer restarts to a Welcome screen and asks you to choose a country or region. If you want to leave the Mac in an out-of-box state, don't continue with the setup of the system. Instead, press Command-Q to shut down the Mac. When the new owner turns on the Mac, the Setup Assistant will guide them through the setup process.

The old machine cannot contain what is on the new machine if you plan not to keep it. It should be wiped and set up as described above.

Their is no such animal as an "extended hard drive." There are external hard drives or internal hard drives. Your Time Machine backup drive is an external hard drive, for example.

You are referring to the Computer Name? Open Sharing preferences and edit the name. If you are trading in the old computer, then you need to perform a factory reset outlined in an earlier post. It details what other things you need to do such as turning off Find My, de-authorize it from iTunes, sign out of your Apple ID and remove it, etc.

There aren't two internal drives, there is only one physical internal drive - Apple SSD - that is partitioned into two logical volumes. Just wipe the drive as part of the factory reset process. This will return it to one physical drive with one logical volume and a version of OS X that came with it.

Unmount or Eject the LaCie volumes, then disconnect it from the old computer. Connect it to the new one.

Yes, you want to do the almost same on the new machine. The new machine you should reset using the Recovery HD. It isn't necessary to do all the clean up that's needed on the old machine.

Install El Capitan or Later from Scratch

If possible, back up your files.

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility from the Utility Menu and click on the Continue button.
3. When Disk Utility loads select the target drive (out-dented entry w/type and size info) from the Device list.
4. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
5. Set the partition scheme to GUID.
6. Set the Format type to APFS (SSDs only) or Mac OS Extended (Journaled.)
7. Click on the Apply button, then wait for the Done button to activate and click on it.
8. Quit Disk Utility and return to the Utility Menu.
9. Select Install OS X and click on the Continue button.

The new machine's SSD should use APFS. I suggest not encrypting the drive, if asked, unless you absolutely must keep your stuff encrypted. Encryption requires using a special password. If you lose it or forget it, then all your files will be forever gone. Write it down somewhere. Then opt for a recovery password to save with Apple, should you lose or forget the main password. Keep in mind that if the SSD gets corrupted, then you may have no access to encrypted files.

Looks pretty normal to me for an APFS formatted SSD using Catalina. I see the two volumes making up Macintosh HD. The LaCie drive is partitioned into two volumes, TM and LaCie. The last entry is a disk image file that is no longer mounted.

So it's all very normal. You have an internal SSD and an external LaCie drive. Now, what do you think is the problem? Is it on this computer (new) or the other one(old)?