The edge, as used most often in my experience, is the entry point that connects the public and private internet. It rarely has anything to do with location (in almost all cases it's implied the edge is regionalized to be the closest data center to the target).
For AWS: a Lambda at the edge, for example, may be a Lambda behind an API Gateway that acts as the public entrypoint into infrastructure behind a VPC. That Lambda is at the edge.
On AWS that’s a somewhat confusing way to put it since there is Lambda@Edge, which allows you to run Lambdas to handle/manipulate CloudFront CDN requests at their edge locations.
For AWS: a Lambda at the edge, for example, may be a Lambda behind an API Gateway that acts as the public entrypoint into infrastructure behind a VPC. That Lambda is at the edge.