The client may do so (and generally would), but the behaviour is not guaranteed (you may find the word
MAY when this is explained in the RFC). Whether it does or not is entirely implementation defined.
The difference between 251 and 551 is that if the SMTP server returns 251 then it will send the mail to the right place as a relay, and the client should not (but still may) send the message that way itself, but if 551 is returned, the client is expected to. This is not a protocol requirement.
If your goal when operating the SMTP server is to ensure the mails are delivered, you should send 251 and forward the mail.
Also, the most current SMTP RFC is RFC5321; generally you should implement SMTP according to that, as there are a large number of features introduced, and a very large part of RFC821 has been removed from the protocol.