Non-Hokey Team Building Activity Ideas

Meredith, I think your ideas are quite good already. Depending on what you perceive to be the reason for the first team to be hesitant to speak up and participate in brainstormings, you could give it just a little twist. If your team is quite diverse then have them also talk about what the think are their personal strengths they bring into the team and then have someone else think about a situation where exactly that strength would be helpful for the team.
For the second team you could include a session to define the team ground rules, not in a general way but with a special focus on the team brand. Like having a weekly retrospective if growth/learning/self-improvement is one of the team values or having monthly presentations from team members for the team on a topic they have special expertise in.
Generally, to break the ice at the beginning of a team building it is also important that the leader goes first in showing vulnerability. I also recommend "The 5 Dysfunctions of a Team" by Patrick Lencioni.