The "ing ..." construction is called a "participle phrase", as denoted by the use of the participle form of the verb. Most often, it is used to indicate simultaneous actions:
The tourists gathered around the lake, taking photographs of the wildlife in the area.
The tourists took pictures of wildlife at the same time as they gathered around the lake.
The cat sat itself down on the veranda, stretching its lazy body.
The cat stretched its lazy body at the same time as it sat on the veranda.
Your revisions change the simultaneous actions to consecutive actions:
The tourists gathered around the lake to take photographs of the wildlife in the area.
The tourists gathered around the lake for the purpose of taking photographs (but no one has actually taken any yet).
The cat sat itself down on the veranda and stretched its lazy body.
The cat sat on the veranda and then stretched (after it was already sitting).
Keep in mind that there are certainly times when you want to avoid the participle construction, because simultaneous action is not
what you want to imply:
Putting on his pants, Jim ran to open the door.
The implication is that Jim is running and putting on pants at the same time, which is probably not the intent of the writer. In this case, a revision similar to the ones you suggest is definitely in order:
Jim put on his pants and ran to open the door.
The two actions happen in sequence, which doesn't leave Jim hopping across the room trying to pull up pants while running. :)