The (US) Bill of Rights is more like
A formal declaration, complaint, or statement of particular things in writing. (Black's Law Dictionary)
See also, a bill of particulars.
I've forgotten, if I even knew, whether the English Bill of Rights was a proper Act of Parliament. The US version is a reference the first ten amendments to the constitution.
But we're talking about
a draft of a proposed law presented to parliament for discussion.
"a debate over the civil rights bill"
synonyms:draft law, proposed piece of legislation, proposal, measure
"a congressional bill"
This is a short-form or casual reference; for example, when reference is made to "the immigration bill", people might mean
"S. 744 [i.e., Senate Bill 744 (of a particular congressional session)]
AN ACT [i.e., a draft Act introduced in the Senate in bill form]
To provide for comprehensive immigration reform and for other purposes."
(When a bill is passed by both houses of Congress and signed by the President, it becomes an Act, which may be referred to as a law or a statute. Pamphlets for civic education are sometimes titled, "How does a Bill become a Law?")