済む and 済ませる: completed tasks and simple solutions
Compare a task being settled with completing it, and examine で済む for a sufficient means or limited burden.
Whether a task is settled
手続きはもう済みました。 says that the procedure has already been completed. 済む, read すむ, refers to the required task being settled, not merely time passing.
支払いが済んだら、領収書を受け取ってください。 asks someone to collect the receipt once payment is complete. 済んだ is the ta-form of 済む; payment comes first.
Complete something needed
出発前に買い物を済ませました。 says that someone finished shopping before departure. 済ませる, read すませる, marks the completed task with を.
昼食をパンだけで済ませました。 means that someone made do with just bread for lunch. It concerns handling lunch that way, not necessarily finishing every piece of bread.
で済む and なくて済む
この手続きは電話で済みます。 means that this procedure can be handled by phone. In a context where a visit is unnecessary, で marks the sufficient means.
友達が迎えに来てくれたので、歩かなくて済みました。 says that a friend's pickup saved the speaker from having to walk. Walking was unnecessary, not prohibited.
Does 歩かなくて済みました mean that walking was forbidden?
Show the answer and why
No. It means things worked out without needing to walk. Distinguish avoiding a necessity from a prohibition.