Sata HDD on new MacBook Pro

another diabolical problem for third-party drives:

Drives like the MyBook came with software, and the manufacturer suggested you install that software.

What that software does is to "spare you the annoyance of having to re-initialize the drive" from its factory set Windows New Technology File System (NTFS, the W is silent because there ARE no other Operating Ssystems) or similar unusual (on a Mac) format. Instead, the software they provided would simulate a MacOS drive inside an NTFS file for your Mac.

The downside of using the manufacturer's software is that if the manufacturer's software is not loaded, the Macintosh Volume may not readable, or may not be writable. This means that in startup manager, Installer, and in Recovery such as after a data loss, the Macintosh Volume may not be visible.

The standard advice given here to all users, including novice users, is to discard the manufacturers software and NEVER use it. "Best Practice" is to erase the physical device when new, using only MacOS Disk Utility, and create the fundamental data structures needed for consistent, reliable use by MacOS.

Be sure to "show all devices" which will allow you to ERASE the entire Physical Device by its immutable manufacturer-given device-name, not just user created Volume-name such as ‘Macintosh HD’.

Many posters complain that an external drive 'doesn't show up' (on the desktop) in later versions of MacOS. the first thing to look at is whether your Finder preferences are set to show them, because the defaults have changed:

and here:

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I saw this happen with a USB stick one time, however, it occurred between two identical Intel Macs with the only difference being the OS running on each one. The USB stick "seemed" to be Ok when used with a single Mac. However, when I could not read the USB stick on the other Mac I also could not easily erase it. Turns out this USB stick was defective with intermittent errors. I never saw anything quite like it before. So it may indicate a possible hardware issue somewhere, possibly with the docking station, the drive(s), or one of the cables or one of the Macs.

You can try checking the health of the hard drives by running DriveDx (free trial period available). You will need to install a special USB driver to check an external drive. However, even with this special USB driver, the docking station may not allow the necessary communication to access the hard drive's health information. For a hard drive any "Warning" or "Failing" notice by DriveDx indicates a bad hard drive which should be replaced (the same is not always true when checking an SSD).

That is a VERY different problem from what you described.

High Sierra sits on the cusp between older MacOS Extended and newer Apple File System (AFFS). High Sierra is the first system to use APFS as the default for its SSD boot drive. However, ROM based software like the Startup Manager was created when your machine was new, and knows NOTHING about APFS drives.

On drives that use APFS, the Recovery Partition still exists, but it is INSIDE the APFS container-disk and in most cases is unreachable on older Macs.