Pretty much! Just a couple extra things to clarify. The first time, you'll actually get 4 pieces of information from ZeroSSL. 1) A Let's Encrypt account key. This is just for you and identifies your LE account. It's helpful to have on hand for revocations and renewals. You can paste this back in when you renew in 3 months. 2) A Certificate Signing Request. Same thing, just paste this back in when you renew to make it easier. Neither of these need to go to your hosting provider.
3) A private key for your certificate. This is important and should be kept safe. You will only get this once, and won't paste it back in. That CSR has the public key corresponding to your private key in it, so when you renew you'll just keep using the same private key. You will need to send this to your hosting provider.
4) the certificate itself. This is not a confidential piece of information, and is the part that expires in three months. You'll also send this to your hosting provider, and when you renew, this is the piece that will change and you'll get a new one of.