"fullchain" is not a type of certificate; it's a commonly used name for a file that contains a PEM-format certificate and its issuing CAs together. (The "fullchain.pem" filename was popularized by Certbot, the original Let's Encrypt ACME client, and is literally a combination of "cert.pem" and "chain.pem" files.)
A full chain file generally looks like this, with the server's own certificate always at the top:
-----BEGIN CERTIFICATE-----
<data of end-entity cert>
-----END CERTIFICATE-----
-----BEGIN CERTIFICATE-----
<data of intermediate cert>
-----END CERTIFICATE-----
-----BEGIN CERTIFICATE-----
<data of root cert>
-----END CERTIFICATE-----
In its typical definition (i.e. what Certbot generates), the "fullchain.pem" file contains only certificates, not private keys – the key stays separate in its own file.
(Note: Although I did include the root CA in the "full chain" example, most of the time you wouldn't actually add the root CA – the client has it anyway.)
(Note 2: Some CAs such as Sectigo call this "reversed"; Sectigo's "normal" chain file is actually root-first and therefore backwards from what TLS requires.)
All certificates are public by definition, there's no such thing as a "private pem cert". The private file only contains a plain private key.
Most likely you can use your private-key file directly without any conversion (assuming your command even did anything at all – in the past it used to be a way to convert PKCS#8-format key files into the older "PEM" format, but now it just outputs PKCS#8 anyway unless you pass -traditional to explicitly request the old format).
PKCS#8 format (the key type is embedded within the data):
-----BEGIN PRIVATE KEY-----
<key data>
-----END PRIVATE KEY-----
PKCS#1 aka "PEM" format (the key type is indicated through the header):
-----BEGIN RSA PRIVATE KEY-----
<key data>
-----END RSA PRIVATE KEY-----
Some programs do require the certificates and the private key to be in the same file; usually in that case the key can just go at the very end. (In very rare cases, though, you need to order them like "host cert → host privkey → CA chain".) But according to Grafana's documentation, that's not the case for you – Grafana expects only the certificates in cert_file, while the key goes in cert_key.