"These are the definitions that the Royal Spanish Academy gives us."
Translation:Estas son las definiciones que nos da la Real Academia Española.
Since the Academy is doing the giving, the verb "dar" needs to be conjugated to "da". Object pronouns generally cannot be attached to the end of conjugated verbs. So, the "nos" has to go before the verb.
Your thinking is correct.
It also seems that Duolingo agrees, since they now accept that word order.
Spanish can sometimes have different word orders depending on the emphasis desired. However, DL does not have a way to let us know one way or the other, so they try to accept both in cases like these.