When my teammates think of me, I want them to see someone respected/important. How can I do that?

For me, there's 2 core things when it comes to quickly gaining respect as a senior engineer:

Showing Your Depth

• As a senior engineer, you should perceive the world differently from junior/mid-level engineers. This can be looking out for more edges cases when building things. It can be from more refined non-technical behaviors like project management and leadership.
• What are you proud of when it comes to your ability to do good work? Figure that out and then show that behavior as much as possible.

Be Kind

• Just because you're new doesn't mean you aren't in the position to help others
• You are naturally in a good position to help junior/mid-level engineers
• Another way to help is to improve onboarding materials for the overall team

Related resources:

• How I Became A Tech Lead At Robinhood 1 Month After Joining - In March 2021, I switched from Meta to Robinhood. This in-depth video covers how I was able to onboard and build a strong reputation quickly.
• Similar question from a senior software engineer at Meta

Adding few more things you can do:

• Volunteer to solve problems
• Ask your manager
• what is 1 thing that I can do to help you or the team in upcoming weeks?

Ask your team members:

• What are the things that slow you down? It can be inefficient oncall or writing same boilerplate code etc.

Once you know these

• go and fix these problems and communicate back to the team, this is one of the best ways to add value and gain respect.
• "Taking initiatives" to make things better is one of the best ways to earn respect.