Postfix change to address based on sending host

Do you need the postfix server to be responsible for example.com? (Do you ever want mail to be delivered there instead of to google?) If you always want mail for local example users to go to their google, remove $mydomain from the mydestination parameter in main.cf. This will not alter where the mails appear to be coming from for outside users. (That is controlled by "myorigin")

Since postfix either thinks it is responsible for the domain or not, there is no easy way to split the domain up so some mail gets delivered locally and some gets delivered to google. You can do some tricks by putting in forwarding rules or with transport mapping, but then you'd need to specify google or local for each user. I kind of doubt you really want to do that.

What you're seeing in GMail is a combination of DKIM and SPF validation. SPF is relatively simple, and is a DNS record specifying which mail servers are designated to send mail from your domain. I believe this is how Gmail presents the "mailed-by" tag. DKIM is a bit more complicated, as you will be cryptographically signing outgoing mail, primarily using a Postfix filter. There is also a DNS entry, to publish the DKIM public key. The DKIM validation is presented in GMail as the "signed-by" tag.

Take a look at here for the official documention on the "signed-by" and "mailed-by" tags.

For a DKIM primer, here's the wiki page. You're using CentOS, but the Ubuntu documentation on setting up DKIM with Postfix is relatively clear. You can search more specifically for CentOS-orientated documentation.

In general, you will be installing the OpenDKIM package (you will need the EPEL repo enabled) and configuring it (e.g., generating the key pair, tell OpenDKIM which key to use when signing a given domain, etc.). There will be an opendkim daemon listening on a port on the loopback interface. Postfix will be configured to process mail through that daemon as a smtpd_milter in main.cf. Once that's done, you will add a TXT record to the DNS for your domain that specifies the public key.