Chamberlan's men is a company
Lord Chamberlain's Men
Lord Chamberlain's Men was created in 1594.
The Lord Chamberlain's Men
Lord Hunsdon's Men became the Lord Chamberlain's Men which became the King's Men.
Shakespeare was an actor and playwright in the Lord Chamberlain's Men who later became the King's Men. He was not in the Lord Admiral's Men.
They didn't change their name--it was changed for them when their patron changed or when his job changed. In fact their name changed several times: From the Lord Chamberlain's Men to Lord Hunsdon's Men when the Lord Chamberlain died and his son Lord Hunsdon took over patronage of the company, from Lord Hunsdon's Men to the Lord Chamberlain's Men when Hunsdon became Lord Chamberlain like his father, and from the Lord Chamberlain's Men to The King's Men when King James I took over patronage of the company. The partners in the company had nothing to say about it.
The name of every Elizabethan and Jacobean acting company tells you the name of the company's patron or sponsor. It's like having an acting company called the Coca-Cola Players. Lord Strange's Men, The Lord Admiral's Men, The Lord Chamberlain's Men and the King's Men were sponsored by Lord Strange, the Lord Admiral, the Lord Chamberlain, and the king, James I, respectively.
Queen Elizabeth's Men, Lord Pembroke's Men, Lord Derby's Men, Lord Strange's Men, Lord Admiral's Men, the Children of St. Paul's, Leicester's Men, Warwick's Men, were some of the rival companies. Not all of them existed at the same time. There were usually four to six at any one time.
Depends how early you want to go. For most of his working life he was one of The Lord Chamberlain's Men, afterwards The King's Men. But he actually did a fair amount of acting before 1594 when the Lord Chamberlain's Men was formed. We cannot be quite sure which company he was with in these early days, but Lord Strange's Men, Lord Pembroke's Men and Lord Sussex's Men are all possibilities.
Although when he was training as an actor, Shakespeare may have appeared as a hired actor with a number of companies (Lord Strange's Men, Lord Derby's Men, Lord Pembroke's Men), he only was a partner in one theatre company, The Lord Chamberlain's Men (renamed The King's Men in 1603), to which he belonged for nineteen years.
the kings men
the king's men