The Rules as Written
All creatures have a walking speed, simply called the monster's speed. Creatures that have no form of ground-based locomotion have a walking speed of 0 feet. Some creatures have one or more of the following additional movement modes.
Long Jump. When you make a long jump, you cover a number of feet up to your Strength score if you move at least 10 feet on foot immediately before the jump. When you make a standing long jump, you can leap only half that distance. Either way, each foot you clear on the jump costs a foot of movement.
High Jump. When you make a high jump, you leap into the air a number of feet equal to 3 + your Strength modifier (minimum of 0 feet) if you move at least 10 feet on foot immediately before the jump. When you make a standing high jump, you can jump only half that distance. Either way, each foot you clear on the jump costs a foot of movement.
Using Different Speeds:
If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speeds during your move. Whenever you switch, subtract the distance you've already moved from the new speed. The result determines how much farther you can move. If the result is 0 or less, you can't use the new speed during the current move.
Your third interpretation doesn't work because of this passage. In effect, you total up the movement used before the change and compare it to the new movement type. You don't track each movement individually, just "consumed this turn" and "total possible in current mode".
The first passage doesn't apply to your example creatures, but untyped movement mode (like the Orc statblock's "Speed 30 ft.") is "walking". If a creature has other types of movement, they will be called out by type. Again, it isn't relevant for your examples, but it's important for parsing statblocks.
The Long Jump and High Jump passages actually supports your Interpretation 1 - a creature without a walking speed is clearly not capable of moving "at least 10 feet on foot" and therefore cannot perform a running jump. It may not be realistic, but we're looking strictly at what is written here.
a dolphin has a strength of 14, a speed of 0 feet, and a swim speed of 60 feet. Could a dolphin swim 10 feet to get a "running start" before jumping 14 feet horizontally out of the water and through the air (subtracting this distance from its swimming speed)?
Dolphins are never on foot, because they don't have feet, and don't have a walking speed. A dolphin on land has a speed of zero. It has no movement to spend, and therefore Dolphins cannot jump on land.
The last passage addresses splitting movement types, this covers your octopus example:
A giant octopus has a strength of 17, a speed of 10 feet, and a swim speed of 60 feet. I would assume it could jump in a manner similar to the dolphin using only it's swim speed - but what if it were on land? Could the octopus use its 10 feet of walking speed to get a running start, and then jump 17 feet horizontally (subtracting the distance of the jump from its unused swimming speed)?
If the octopus is on land, it must use walking movement. It has limbs that can serve the same purpose as feet, so we'll give it a pass on that threshold, too. As both types of jump need ten feet of movement before a running jump, an octopus cannot do any sort of running jump - they're out of walking movement after ten feet. That said, an octopus still has a walking speed, Strength 17 allows for a standing long jump of 8.5 feet, or a standing high jump of 3 feet.
What D&D's rules are primarily concerned about is useful tactical movement in a combat situation. Anybody who has been to Sea World knows that dolphins most certainly can jump out of the water, and when they're on land they can sort of waddle-wiggle from dry land into the water. Neither of these would be terribly useful in combat, and submarine-launched ground attack war dolphins aren't a thing (as absurdly amusing as the image may be).
As far as water-to-water jumps go, or perhaps jumps out of the water, I would allow a creature that started in the water to use swimming speed to meet the 10-feet pre-move requirement for running leaps, and subtract the jump distance from that swimming speed. What they can do after the jump depends on where the creature lands and what available movement modes it has. This where the last passage comes into play.
- If the dolphin swims 10 feet, then jumps for 14 feet, it has used 24 feet of movement. If it lands in the water, it can keep going because 60 - 24 = 36, greater than zero. If it lands on the ground, it's done moving because it has to "switch modes" and 0 - 24 is definitely less than zero.
- If the octopus swims 10 feet, the jumps for 18 feet, it has used 28 feet of movement. Landing in water leaves it with with 32 feet. Landing on the ground also leaves it with less than zero walking movement. It's done moving for the turn, but can walk on its next turn.
Some of that jumping movement would be "over" land, but I would still count it against swimming speed until the creature actually lands.
Going the other direction, a dolphin on land is out of luck. It has no walking movement speed and is pretty well immobilized. The octopus on the other hand could potentially leap an 8.5 foot wall, land in the water, and swim another 51.5 feet (or be dumb, swim 10 feet, jump 18 feet back onto land, and be stuck for the rest of the turn).
This is most similar to your Interpretation #2, but takes Using Different Speeds into account.
As an aside, partially because it's irrelevant to 5E and partially because it was a semi-traumatic experience, the 3.xE-era OGL setting/system Blue Rose had uplifted animals available as player characters. Dolphin was an option, and it could do a full round action (a 3.xE term for "do nothing else the entire round") to move 5' on land. It also had rules for skill checks without appropriate hands, so suffice a phrase that circulated in my playtest group was "No, Flipper! Not the forge!"