Syslog driver not catching error logs properly

Description

Hello,
When I try to use syslog as a log driver, Docker sends all messages logged inside the container as INFO. Therefore I cannot catch errors/warnings in syslog coming from docker containers. Logs forwarded to stderr should be treated as errors and sent to syslog as ERROR priority so they can be caught by monitoring tools.

Reproduce

docker run --rm -ti --name test-log --log-driver syslog --log-opt syslog-facility=daemon --log-opt tag=docker --log-opt syslog-address=unixgram:///dev/log debian bash -c 'echo "stderrDOCKER" > /dev/stderr'

Expected behavior

No response

docker version

Client:
 Version:           20.10.24+dfsg1
 API version:       1.41
 Go version:        go1.19.8
 Git commit:        297e128
 Built:             Thu May 18 08:38:34 2023
 OS/Arch:           linux/amd64
 Context:           default
 Experimental:      true

Server:
 Engine:
  Version:          20.10.24+dfsg1
  API version:      1.41 (minimum version 1.12)
  Go version:       go1.19.8
  Git commit:       5d6db84
  Built:            Thu May 18 08:38:34 2023
  OS/Arch:          linux/amd64
  Experimental:     false
 containerd:
  Version:          1.6.20~ds1
  GitCommit:        1.6.20~ds1-1+b1
 runc:
  Version:          1.1.5+ds1
  GitCommit:        1.1.5+ds1-1+deb12u1
 docker-init:
  Version:          0.19.0
  GitCommit:

docker info

Client:
 Context:    default
 Debug Mode: false

Server:
 Containers: 0
  Running: 0
  Paused: 0
  Stopped: 0
 Images: 4
 Server Version: 20.10.24+dfsg1
 Storage Driver: overlay2
  Backing Filesystem: extfs
  Supports d_type: true
  Native Overlay Diff: true
  userxattr: false
 Logging Driver: json-file
 Cgroup Driver: systemd
 Cgroup Version: 2
 Plugins:
  Volume: local
  Network: bridge host ipvlan macvlan null overlay
  Log: awslogs fluentd gcplogs gelf journald json-file local logentries splunk syslog
 Swarm: inactive
 Runtimes: runc io.containerd.runc.v2 io.containerd.runtime.v1.linux
 Default Runtime: runc
 Init Binary: docker-init
 containerd version: 1.6.20~ds1-1+b1
 runc version: 1.1.5+ds1-1+deb12u1
 init version:
 Security Options:
  apparmor
  seccomp
   Profile: default
  cgroupns
 Kernel Version: 6.1.0-21-amd64
 Operating System: Debian GNU/Linux 12 (bookworm)
 OSType: linux
 Architecture: x86_64
 CPUs: 4
 Total Memory: 15.62GiB
 Name: vpc-ret-fv-001
 ID: QNS6:QCAY:O74G:SFND:ERNG:4F2A:YGSB:Z6C4:PYFR:VMTL:JCL4:KXR7
 Docker Root Dir: /var/lib/docker
 Debug Mode: false
 Registry: https://index.docker.io/v1/
 Labels:
 Experimental: false
 Insecure Registries:
  127.0.0.0/8
 Live Restore Enabled: false

WARNING: bridge-nf-call-iptables is disabled
WARNING: bridge-nf-call-ip6tables is disabled

Additional Info

No response

The issue provides a reproducible docker run command but names no source files or tests. Start by reproducing the syslog driver behavior with stderr output, then trace the Go implementation that maps container streams to syslog priorities. Done means stderr messages are emitted with ERROR priority while the existing driver behavior remains intact.