Is short-term debt the same as current liabilities?

Essentially, yes.

Many times a company has Long-term debt, with a certain amount to be repaid within the year. On the company's balance sheet they will have the remaining amount of their Long-term debtincluded in Non-Current Liabilities, while in Current liabilities they will have the Current portion of long-term debt.

Basically, the balance sheet has a section for Current liabilities, which would include accounts with debts to be repaid in the short-term (generally within the year). Normally it is not listed as Short-term debt, but rather an account like Accounts payable or Bank loan, or as I stated earlier, Current portion of long-term debt.