Including Id in URI for PUT requests

I was reading some documents about the appropriate use of URI's using rest services and I came across an example for basic GET .. DELETE requests.

The example uri's were:

Get all users

GET http://mydomain.org/api/users

Get specific user

GET http://mydomain.org/api/users/1

Update user

PUT http://mydomain.org/api/users/1

DELETE user

DELETE http://mydomain.org/api/users/1

A user resource would be either JSON or XML in the form of:

{
    Id: 1,
    FirstName: 'John',
    LastName: 'Doe'
}

My question is this. To maintain REST principles, is it required to include the id of the resource within the URI for PUT requests?

The PUT method requests that the state of the target resource be created or replaced with the state defined by the representation enclosed in the request message payload.

You want to PUT a resource to the same URI you intend to GET it from.

RFC 72314.3.4 PUT

I was going to ask a similar question, but I think I found the answer. I am not sure if it is by REST principles, but here is why it would be bad not to include ID in the URI. So say your PUT is like:

PUT http://mydomain.org/api/users

And then you happen to update multiple users with different id's but the same URI cause there is no ID in your URI. Then, an important thing to know here is that PUT is idempotent http verb. This means that calling it once should have the same effect as calling it mulitple times. Therefore, some intermediate node in the network, just following the fact that you PUT multiple times, might ignore all but one of your requests cause they have the same URI. Finally, that's definetely not what you want cause the intention was to update multiple users not only one.