client and server ssl profiles

client (side) profile means config to be used by F5 when communicating with client.
server (side) profile means config to be used by F5 when communicating with server.

you should put the CA-authorized certificate in F5's client profile.
if your wild card certificate is CA-authorized, not self signed, then you should put it in the client profile.

for server side profile, usually you can simply use the default server profile.
and the webserver can also use self signed certificates.
i usually configure weaker ciphers for server side profile, e.g. AES128 instead of AES256, to reduce server load.