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If I get a bill (e.g. for internet service), is that a debt I owe? If no, what are the practical difference between a bill and a debt?
From accounting perspective, an unpaid bill for internet services, according to the Accruals Concept, is recorded as a liability under 'Current Liabilities' section of the Balance Sheet. Also as an expense on the Income Statement.  So to answer your question it is both: a debt and an expense, however this is only the case at the end of the period. If you manage to pay it before the financial period ends this is simply an expense that is financed by cash or other liquid Asset on the Balance Sheet such as prepayment for example. For private persons you are generally given some time to pay the bill so it is technically a debt (Internet Provider would list you as a debtor on their accounts), but this is not something to worry about unless you are not considering to pay this bill. In which case your account may be sold as part of a factoring and you will then have a debt affecting your credit rating.