Existing and error logs behavior (#4656)
### Description
- Adds an integration test for existing behavior (`continue.t`)
- Fixes behavior of `--output-logs=errors-only` to match behavior for
`full` in the case of a task that errors
- Fixes `--continue` behavior to report all failed tasks at the end
### Testing Instructions
New `continue.t` integration test. I recommend viewing the changes to
that file at each commit to see initial behavior and how it changes.
Fixes #4504
link WEB-895
