The Congress met in the same room the Constitutional Convention would use eleven years later, with the windows shut for secrecy, in a Philadelphia summer. It had no legal authority, no power to tax and no army until it invented one, and it sat more or less continuously for six years.
Three separate things, three separate dates
Richard Henry Lee moved for independence on 7 June. The vote was deferred for three weeks so delegates could get instructions from their colonies — several were under standing orders not to vote for separation and had to write home for permission.
On 2 July, twelve colonies voted yes and New York abstained. That was the decision. On 4 July, Congress approved the text explaining it, after two days of line editing that cut roughly a quarter of Jefferson's draft, including his passage attacking the slave trade.
John Adams wrote to his wife that the second of July would be celebrated by future generations as the great anniversary festival. He was off by two days, because the date printed at the top of the document is the one everybody kept.
The signing was not an event
Most delegates signed on 2 August, when the engrossed parchment was ready. Some signed later still; at least one signer had not been a member of Congress on 4 July. The scene of fifty-six men queueing at a desk on Independence Day did not occur.
Trumbull's painting, which hangs in the Capitol rotunda and is on the back of the two-dollar bill, shows the drafting committee presenting the text — a real moment, on 28 June — with rather more people in the room than were actually there, because he added portraits of signers he could get sittings from.
What the Congress was really doing
Independence is the memorable output. The unglamorous one is that this body ran a war for six years by consensus among delegations that could be recalled by their own colonies at any time, with no executive, no reliable revenue and communications measured in weeks.
It worked badly and it worked. The Articles of Confederation that came out of it were weak enough that the same building had to be booked again in 1787.
What people get wrong
Nothing much happened on 4 July 1776 except the approval of wording; the decision was taken on the 2nd, and the signing was in August. The Liberty Bell was not rung to announce it — that story appears in a work of fiction in 1847. And the parchment everyone pictures is the engrossed copy made after the fact; what Congress actually approved on the 4th was sent to a printer overnight, and about two dozen of those broadsides survive.
Adams thought the second of July would be the anniversary. The date at the top of the document won.