The two popular words for this would be "dive" (or "dove", past-tense) or wade
To "dive into the water" is close in meaning to saying "jumping into the water so as to entirely (or mostly) cover oneself in water". But to "wade into the water" generally means a slower and more gentle entry into the water, and to be incompletely submerged in the water. To wade in the water further implies walking rather than outright swimming.
So if you want to note going into the water and being completely submerged, you might say "dive under the water" instead of "dive into the water", as this makes it entirely clear that you mean complete submersion - but either is pretty clear.
Generally, when "stand" is used to mean "is in a standing position", it is accompanied by a location:
He stood in the corner, sipping a whiskey.
They stood around the table, arguing.
She stood by the window, looking forlornly down the empty driveway.
Often this meaning will be expressed in the past continuous, using "was standing" instead of "stood" to unambiguously refer to the condition of standing, as opposed to the act of standing:
He was standing in the corner, sipping a whiskey.
When "stand" is used to mean "get to one's feet", it is either unaccompanied, or accompanied by an event:
He stood when she entered the room.
They stood as the speaker finished his speech.
"Now that you're here, I can finally leave," she said as she stood.
In my experience, stand up is rarely used for "be in a standing position" when it refers to a person; you are much more likely to hear it used for an inanimate object (especially one which is normally horizontal), and again it will most likely be accompanied by a location:
The bedframe stood up against the wall.
Most often, stand up is used for the action of getting to one's feet:
He stood up and stretched
As I approached, he stood up and reached to shake my hand.
Everyone stood up when the orchestra finished its song.
In your first two examples, people changed from sitting to standing when the president came in, and people changed from sitting to standing for the opening of the speech (they did not remain standing for the entire speech, but sat down again after only a moment or two).
Your given alternatives, "get to one's feet" and "rise", both unambiguously refer to the action of changing from a non-standing position into a standing position.
Best Answer
Crouching is different from "bending down." The picture that you have provided is of someone crouching, with his center of gravity in the same vertical plane as it would be were he standing. A person who is bending down would bend be leaning forward with his center of gravity ahead of where it would be were he standing.
To "duck" might imprecisely mean to crouch very quickly, but it could also mean to bend over very quickly or to fall to the ground quickly.