Mr Yedhula - very well written explanations on the topic. But I beg to differ a bit on some issues (my remarks given in bold underline):
A usual guarantee is governed by Sec. 126 of the Indian Contract Act, 1872. A bank guarantee is not directly governed by Sec. 126. --- All guarantees are covered under Sec 126. Bank Guarantees (BGs) are called so because they are issued by Banks. Similarly, guarantees issued by individual and corporates are called Personal Guarantees and Corporate Guarantees, respectively.
An ordinary guarantee is a tri-partite (3 parties) agreement involving the surety, the debtor and the creditor. But a bank guarantee is a contract involving two parties i.e. the bank and the beneficiary --- Guarantees are NOT tri-patrite agreements. Guarantees are contracts and not agreements (guarantees are executed/signed only by the guarantor). Yes - 3 parties are involved in all guarantees, including BGs. In BGs, the 3 parties are BG applicant (bank customer), Guarantor (the Bank), and the Beneficiary.
If my remarks are wrong, please enlighten me and others.