Yes, we are saved by our faith.
Deeds done without faith, cannot be deeds done to please God, but with some other intention.
A person with faith needs to
ACT on that faith.
Hence, we have in the Qur'an "those who believe and do good works, will inherit paradise"..
Why should a believer who couldn't care less, be treated as a believer who strives to please his Lord?
..and we also have "whoever has faith, even as small as a mustard seed, will eventually enter paradise".
In other words they might enter hell first, but
eventually come out .. much like the Catholic purgatory.
Much like your definition, I would think. Belief in God, and the Scriptures.