tl;dr: Had inputted information incorrectly and You have to add AWS API Gateway to the list of users that can use AWS Lambda, this is done by creating a user role, giving the role permission to access use AWS Lambda, and adding AWS API Gateway to the list of trusted users that can use the role. This can all be done using the AWS IAM web portal.
Hey, I got it to work, but I am unsure as to why I could see some lambda functions and not others.
For integrating the lambda function as an AWS resource using the AWS API Gateway web interface: I was entering the lambda ARN in place of the access policy ARN.
For integrating the lambda function as a lambda function using the AWS API Gateway web interface: In the IAM settings I had a role that had permissions to access lambda functions but I had not added API Gateway to the trusted list of users. When I updated the policy, adding AWS API Gateway to the list of trusted users, I could see newly created lambdas.
I don't know if these changes are what fixed the problem, but it works now. Previously AWS API Gateway had been added to the list of trusted policy users. I removed it when I was not going to be using lambda functions. After updating the policy today, I could see newly created lambdas. Before updating the policy today, I had been only able to see lambda functions that the API gateway could access before I had originally removed the policy. Also, for the duration of time that the API Gateway was not added to the list of trusted users, some of the lambda functions that you could view through the API Gateway web interface were listed as being in different regions then where they currently were, or had ever been; there were even some that I do not think I had added.