Sorry. That's you're interpretation It never makes the distinction btwn the God of the Bible and that of the Qur'an, but after the "profess" part it goes on to affirm they are the same God:
“The Moslems together with us adore the one merciful God.” (http://www.vatican.va/archive/hist_councils/ii_vatican_council/documents/vat-ii_const_19641121_lumen-gentium_en.html)
"Together" means together, and their is no pleading that this was akin to the unknown god of Acts 17, as it has a distinct theology.
Instead it is the result of doctrine by a committee of competing schools, both of which subject Scripture to being their servant.
Time for sleep now.