You have it correct. You'll need the SSN to be issued by October 15 or the child will be unable to be claimed as a "qualifying child" on your 2020 return and you won't get the credits or stimulus for them. They have to be a qualifying child in order for you to qualify for stimulus for them.
Here is a link from the IRS explaining this rule: https://www.irs.gov/faqs/filing-requirements-status-dependents/dependents/dependents-9
Even if you don't get the SSN issued by October 15, you might still be able to claim the third round of stimulus for the child on your 2021 return if the SSN is issued in time to file that return.
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