Update: I believe I have found the answer. Posting here for anyone else who needs to do the same:
In my case, since I am using MDDHosting's SMTP servers to send mail while sending them from the Gmail UI, I don't need to configure Google's SPF domain name in my SPF record config. However, on the Email Authentication page in cPanel, I had to add the IP block for the outbound smtp server that MDD uses. In my case it was 22.214.171.124 (outbound.mail.supportedns.com).
Once I did this, to confirm it was working, I sent and email from domain's email to my gmail ID and looked at the Message Headers (Show Original). There you should see something like this:
spf=pass (google.com: domain of firstname.lastname@example.org designates 126.96.36.199 as permitted sender) email@example.com.
In case you do send mail from Gmail and use Gmail's SMTP servers instead to forward emails (in my opinion, you shouldn't do this if you want to look professional to your clients), this link describes what you need to do with your SPF record: https://support.goog...a/answer/178723
To be extremely reserved, you can modify the SPF record to end with -all instead of ~all. Do this only if you understand the consequences.
Hope this helps.