Which OS will run faster on MacBook Air 2014 / 128 GB HD / 4 GB RAM

Mavericks is no longer available for download, it may still be available from Apples Recovery Servers.

I would start with Mac OS X El Capitan (10.11.6), you can download it here.

How to upgrade to OS X El Capitan – Apple Support

Go to Section 4 and click on Download OS X El Capitan.

This downloads InstallMacOSX.dmg to your Downloads folder.

The next section can only be done on a mac that is capable of running El Capitan,

a mac that came preinstalled with an OS later than El Capitan will refuse to do the next bit.

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

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

but converts the InstallMacOSX.pkg to the Install OS X El Capitan.app which 

you will find in your Applications folder, it should be 6.2GBs in size.

(If the installation window asks which disk you want to install to, you must pick 

the disk that you are booted to at the time. Not any internal or external disk that 

you want to eventually install El Capitan on, that is for later.)

Create a bootable USB installer disk using the Install OS X El Capitan.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

After that you can consider upgrading further, I would recommend going to macOS High Sierra first

as this should allow you to download and install the firmware necessary to convert the file system

from HFS+ to the newer APFS which the modern OS's are optimised for.

Please use Safari to access the following links do not use Firefox

How to upgrade to macOS Mojave - Apple Support

How to upgrade to macOS High Sierra - Apple Support

Before upgrading make a full back up of your Mac.

Check that third party apps you use have updates available for the new OS.

https://roaringapps.com

Check that your printers/ scanners etc have updated drivers and software available for the new OS.

Connect your mac to your router via cable rather than rely on WiFi.

Do not use your mac while it is downloading

and do not use any other devices that share the same internet connection.

If the App Store says ‘Open’ then it is detecting a previous download of the install app

go to the Applications folder locate Install macOS ******.app.

Move it to the Trash, empty the Trash.

Restart, click on the link again, the App Store should say ‘Get' or 'Download'.

If you are considering upgrading to Catalina be warned that it does not support 32bit apps.

Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,

some updates may be free, some may be paid. Use this app to discover which apps you have

are 32bit, https://www.stclairsoft.com/Go64/. Once the list has been created you can then

contact the developers of the apps to see if they provide 64bit updates. 

How to upgrade to macOS Catalina – Apple Support

You can also download High Sierra, Mojave or Catalina to install on your 2014 MBA.

Although 4GBs RAM is a bit low for Mojave and Catalina.

Use Safari, Firefox will not work and click on the links below.

How to upgrade to macOS Catalina – Apple Support

How to upgrade to macOS Mojave - Apple Support

How to upgrade to macOS High Sierra - Apple Support

Whichever one you decide to download the Install macOS ******.app will be

downloaded directly to your Applications folder.

Then you can make a bootable USB installer using the instructions here,

How to create a bootable installer for macOS – Apple Support

You should never wipe your mac and then try to recover from Apples Recovery Servers.

Before wiping your mac you should have a copy of the OS you want to reinstall on

a bootable USB installer, as described above.

The certificates for several of Apples OS's expired in October 2019, Apple have been very poor at getting these updated on their Recovery Servers with valid certificates. This may be the problem you are experiencing. 

Try this workaround, we are going to set the time and date on your mac to a date previous to the expiry date of the certificates.

Connect your mac to your router via cable, not WiFi

Boot to your Recovery HD, click on Utilities in the menubar select Terminal.

Make sure WiFi is switched off, it can reset the date back to today.

Enter a new date, for example or just copy and paste

sudo date -u 011421002016

press Return

enter your password

press Return

If Terminal returns an error saying sudo : command not found, then try again without sudo.

Now try downloading the OS.

If this works then when the OS is installed and booted up you can Open System Preferences> Date & Time

and reset the time back to today.

Are you sure you are following the instructions for resetting the date properly, for example not using WiFi

and making sure WiFi stays switched off.

As I said if your other mac came preinstalled with an OS later than El Capitan it will not do the procedure to convert InstallMacOSX.pkg to the Install OS X El Capitan.app.

So older macs that can be upgraded to El Capitan but haven't can do this. Macs running El Capitan can.

And any macs that at one time may have ran El Capitan but have been upgraded further to any of the newer

OS's can do the conversion.

APFS is only for use on the internal disk of your mac. Time Machine can only be run on Mac OS Extended (Journaled)

disks. Apple have made sure this works. Why you have chosen Mac OS Extended (Encrypted) I don't know.

There are places that will sell you El Capitan on a bootable USB, they may be riddled with malware and/

or be the older expired version.

As was been made clear in my first post which you quoted back in a reply to me , a mac that came preinstalled with an OS newer than El Capitan will refuse to do the conversion.

An MBP from 2018 would have come preinstalled with macOS High Sierra or if a later 2018 model macOS Mojave.