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Broadly speaking - unless you are a pro, you can. On ebvry tee, and whenever the Rules allow for a change of balls.Only in tournaments with the "one ball Rule" in effect, you can't.
So, if I hit a ball in a lake I can pick a different make of ball entirely for my next shot (assuming I don't wade in and find it).
If you hit your ball into a water hazard you may take a new ball, even if you can reach the first one. Rule 26-1 allows this.
As do Rules 27 (OoB) and 28 (Unplayable).
...And Rule 25-1 if the ball is not easy recoverable - the player may take another ball into play, if the ball is lying in a deep pool of casual water, visible but hard to reach.
Can you use a different ball, in the instances you mentioned, if the one-ball rule is in effect.
If the one-ball Rule is in effect, the player must use the same type for every stroke. As far as I can recall Parnavik had to leave a tournament once, because he ran out of balls, and nobody else was playing with the same brand.(Could be a myth, though)
Tiger Woods nearly came to the same fate in 2000 when he was leading the US Open at Pebble Beach and went on to win it.The night before he had taken three of the five balls Steve Williams always carries in the bag, to practice his putting on the hotel room floor and forgot to replace them.On one hole using the driver he smashed it into the rocks. Only then Steve Williams realised he had one ball left. Williams says he saw his career about to come to an end if Tiger lost the new ball he gave him.He then tried to convince Tiger to hit a 2 iron instead (without telling Tiger why), but Tiger was furiuos with his wayward drive and you can actually lip read him saying to Williams, "..give me the *&^%ing ball". Of course he smashed it straight down the middle and only learnt of the tightrope he was walking at the end of the round.
Rules DocI remember something like this this happening and a little bit of digging found it was to Derek Cooper. He hit five balls in to the water on the 16th at the 1998 Scandinavian Masters which was won by Jesper Parnevik (which may explain the link)