You get the public key from your private key.
The generator only accepts compressed public keys.
Create a Compressed WIF Private key store it securely as you store any other private key,
a wallet can derive the public key from the private key.
Make shure your device is not compromised by malware, best practise is to do this Offline!
Use an existing wallet or create a new wallet with a seed and strong password.
Backup the seed and the wallet.dat!
Click on "View" -> "Show Addresses" -> Click on the Button "Addresses" at the header,
choose one of the addresses and Label it e.g. "key for vanityaddr"
Get the Public key:
Click with the right mouse button on this labeled address again and choose "Details",
it shows the public key (WIF compressed).
Copy the public key or scan it as QR-Code and paste it here!
Get the Private key:
-> click with the right mouse button on this labeled address and choose "private key", by entering your password it shows the private key beginning with "L" or "K",
you can show it as QR-Code to import this private key into your favorite mobile wallet. You need this Private key later for the key merging.