So, according to the tags you placed on your question, you're using Ruby on Rails.
I have the impression you have two schedulers running (one was initialized 2 seconds after the first one, as hinted by the log output).
You can easily place something like
log "started Scheduler #{scheduler.object_id}"
after the scheduler initialization to see what's going on.
You don't mention what you run Rails on top of (Webrick, Passenger, Thin, Unicorn, ...?). Some of those run multiple copies of the Rails process.
You could use the locking system in rufus-scheduler to prevent a second (or a third...) scheduler from starting.
For a simple locking mecha: https://github.com/jmettraux/rufus-scheduler/#lockfile--mylockfiletxt
If you need more (running multiple Rails on different hosts maybe): https://github.com/jmettraux/rufus-scheduler/#advanced-lock-schemes
The variations you see in your EDIT are probably tied to the server forking new Rails copies to cope with demand.