Q: Sierra installer problem to USB HD

I now have Sierra installed on the external usb HDD and I have one further question and a problem.

Question:  When I start up with the option held down the computer offers me startup options from the internal HD.  I do a 'hard' shut down and start up again with option held and I get the startup options from the internal and the external HDDs.  Why?

The problem:  Sierra on the external drive can start my MBP but not the Air, which is the one I want to wipe and do a fresh install.  Any suggestions please.

Thanks for your response and I'm sorry for the complexity of the problem.  It's a 13" mid 2012 MacBook Air.  The user account has been migrated through countless machines possibly as far buck as 10.0 or 9.2 if that is possible.  I am getting occasional App crashes, even with simple stuff like Acrobat Reader.  I therefore wanted to start the computer up from an external disk, wipe the internal HD and make a fresh OS intallation.

If you don't mind I would like to work forward from the current position.  At the moment I have Sierra installed on an HDD in an external USB caddy.  It can start my MBP8,2 but when connected to the AirBook the startup hangs after a minute or two and the display shows a circular symbol with a diagonal line across ('road block'). 

I tried to start up in diagnostic and hardware test mode but it hung after connecting to the wifi.  If I do a normal restart the Macbook Air starts up fine.

Thanks for the suggestion lllaass.  My main concern with your approach is that there is a hardware problem and that I might hang half way through an installation and then my wife won't have a working computer next week.  The MBA won't boot off the external HDD Sierra:  I suppose it won't hurt to try booting off a bootable installer.  I'll try it and let you know how it turns out.

If it was my MBP I would clone the disc and swap out the HDD if there was a disaster, but with these fancy thin-edged boxes its a different matter.  IMHO lack of versatility is a high price to pay for aesthetics, but that's just me moaning.  

Thanks, Diskmaker is a nice little app that saves a bit of typing on the terminal.  Unfortunately, I did not get far before I received this message

"The disk could not be created because of an error: An error occured: -10006. Finder got an error: Can’t set disk "Install OS X Yosemite" to "DMX_Workdisk"."

I had already tried it with the sierra installer and got the same message.  I am working on the Airbook.  I think the USB caddy is working fine because I have used it to move stuff between three computers and it all works fine.  It is something about this Airbook that won't let it create a bootable USB or be booted from an external USB.

BTW, its not the tear down that bothers me, it's the fact that the SSD is not a standard 2.5 inch, which are cheep and lying round the place ready to grab.

Next approach it to make the bootable install disk on my iMac and then point it at the Airbook.  I'm off for dinner and will let you know later.