Mac will boot from Sierra installed on external hard drive, but how do I get it to install to the laptop and boot on its own?

Disconnect that drive then boot your Mac holding down Option.

You will see a hard drive icon. Does it say recovery ? Does it say Macintosh HD?

If you see Macintosh HD, hold down the Control key then click the circle arrow.

That will set the startup disk as Macintosh HD.

If that doesn't work, just reinstall from internet recovery.

https://support.apple.com/en-us/HT204904

Ok with the external drive connected, boot your mac up. Now open Disk Utilities (Applications/Utilities)

Select your internal drive in the left column, make sure you select the Disk not the indented Volume.

Click on Erase, give the Disk a name, at Format select Mac OS Extended (Journaled) and at Scheme

select GUID Partition Map, click on Erase. Once finished your internal disk should now be okay to install an OS on.

Quit Disk Utility.

Click on this following link, How to upgrade to macOS Sierra - Apple Support

go to Section 4 and click on Download macOS Sierra.

This downloads InstallOS.dmg to your Downloads folder.

When downloaded open to InstallOS.pkg, double-click on

that and an installation window will open, this does not install Sierra

but converts the .pkg to the Install macOS Sierra.app which you will find in 

your Applications folder, it should be 5.04GBs in size.

To start the installation of Sierra double click on the Install macOS Sierra.app.

The copy of the install app self deletes after installing Sierra, so make sure you keep a copy of the InstallOS.dmg

if you need it in future, or you could just make a copy of the Install macOS.app prior to installing and moving it to

an external drive for safe keeping. It is also possible to create a bootable USB installer disk, using the Install macOS Sierra.app

in the Applications folder, and the createinstallmedia command in the Terminal app. Read the instructions here,

How to create a bootable installer for macOS – Apple Support

The download of Sierra was released in October 2019, after the previous installers certificates expired, so this is an updated version. When it was first released there was a problem with it, it would not create a bootable USB installer. I am not sure if

Apple has bothered itself to rectify this, but try it any way.

Your mac can install High Sierra and you can download it from here,

How to upgrade to macOS High Sierra - Apple Support

At Section 4 click on Get macOS High Sierra, this downloads Install macOS High Sierra.app

directly to your Applications folder. The installer is about 5.23GBs.

If you do not want to install High Sierra you could install El Capitan, it downloads the same way as Sierra so use the

above instructions for Sierra above. How to upgrade to OS X El Capitan – Apple Support