Including unrelated punctuation in the quotes

I have seen in many places, in many articles on the web as well as in print, the practice of including punctuation inside the quotes, when the punctuation is not related to the quoted content, as follows:

"We have begun development," the spokesperson told the media.

Although his response was, "We have begun development," he could not
share any further details at this time.

The comma at the end quote is not related to the quoted content but it is included in the quotes. Why is that?

Shouldn't the following be correct, with the comma outside the end quote?

"We have begun development", the spokesperson told the media.

Although his response was, "We have begun development", he could not
share any further details at this time.

The next example is about the full stop at the end. This is included inside the quotes.

"We have begun development, " the spokesperson said, "but I cannot
give you any further details at this time."

Here there are two functions to accomplish, marking the end of the quoted content as well as the end of the sentence, but only one full stop is used.

Does the full stop, in this case, mark the end of the quoted content, the end of the sentence or both of them?

Is the following correct, marking both with different full stops?

"We have begun development, " the spokesperson said, "but I cannot
give you any further details at this time.".