During the recently-ended 116th Congress, H.R. 6192 was approved by the Senate on Dec. 17th, 2020, and presented to the President on Dec. 24th. The 116th Congress adjourned for good on Jan. 3, 2021 with H.R. 6192 unsigned. By my understanding of the law this should constitute a Pocket Veto, and the Bill should be officially dead. However, President Trump signed it on Jan. 5, 2021, the 10th day (excluding the day it was presented to him and excluding Sundays) since it was presented to him. Is this now a Law, or did the Bill legally die on Jan. 3, 2021?
The US President has 10 days, not counting Sundays, to sign a bill, thereby enacting it into law, even if Congress has adjourned. If he neither signs nor returns it, and Congress has adjourned, it does not become law. This is known as a "pocket veto" as the president is said to simply "put it in his pocket" and forget about it. It differs from a regular veto in that the President need not provide reasons for not signing the bill.
While Congress is in session, if the President does not sign a bill within 10 days, but does not return it, it becomes law just as if it had been signed.
H.R.6192 - 1921 Silver Dollar Coin Anniversary Act became Public Law No: 116-286 on January 5, 2021. Clearly, it wasn't vetoed.
From a comment on the question,
But the old Congress ENDED on Jan. 3, 2021. The President did not sign the Bill until Jan. 5. Does an unsigned Bill expire when a Congress ends, or not? – CaptHenway
He may sign within ten days (Sundays excepted) after the bill is presented to him, even if that period extends beyond the date of the final adjournment of Congress.
On one occasion in 1936, delay in presentation of a bill enabled the President to sign it 23 days after the adjournment of Congress.
In the US, a pocket veto can only occur where Congress submits a passed bill to the President and then adjourns during the ten-day period after the bill is submitted. The rationale for this is that a bill must return to Congress with the president's veto within that period so that they can try to overturn the veto, but since an adjourned Congress cannot receive a bill or act on it, the bill is simply voided.
Congress can adjourn anytime it likes within its session, but Congressional leaders can also recall members at any time, so using a pocket veto requires a bit of political calculus. Presidents might use a pocket veto, for instance, on bills submitted before Congress goes on holiday when they are certain that there isn't enough support in Congress to override the veto, on the understanding that Congressional leaders will not pull members back from holiday for a losing vote. That way they can tacitly veto a bill without publicly signing their name to its defeat. Likewise, Congress may choose to send bills they know the president wants to veto when there is a political advantage in being seen to pass a bill but have the bill die more-or-less silently. The only time Congress cannot forestall a pocket veto by recalling adjourned members is in the ten-day period before a new Congress is sworn in (every two years, after some portion of Congress faces elections, and new Senators or Representatives must be sworn in). When a new Congress is sworn in, all unsigned bills are voided.
Any bill that has been vetoed — whether by a normal veto or a pocket veto — can be passed by Congress again and resubmitted to the President, who could then decide to sign it. This isn't a normal procedure by any means. A veto usually signals that a president is unwilling to sign a bill as it stands, and that Congress must either override the veto by two-thirds vote, scrap it, or rework it and submit revised legislation. But this is merely a procedural matter, not a hard and fast rule. For instance, if Trump had pocket-vetoed the National Defense Authorization Act, and then had won the election so that he continued into a second term, the new Congress would almost certainly have resubmitted the bill with no alterations as their first act, and then Trump would almost certainly have signed it, knowing that a veto would be decisively overturned by Congress.