Managing balances sounds like loans to me.

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Kinda. Here's how I understand it. We each put $5 in a jar. I then buy something from you for say $3, but we say we'll just figure it out later. You buy something from me for $2, same. We decide to settle up and you take $6 from jar and I take $4. Take this and multiply by all the transactions and hook a bunch of jars together.

Contract is a better word. If you have 1m sats, you create a channel with that amount, which is basocally a contract between you and your peer that says "I have 1m and you have 0."

A LN transaction is a balance-update to the shared contract, using cryptography to ensure both sides agree atomically on what the lateat balances are