Suppose I have two black boxes that look and feel the same to me. I push down on the top of each one and it goes down. Let each black box and its contents be a system, more specifically, a control mass. In each case, I have increased the energy of the system. By mechanical means, I have caused energy to flow into each system. I say I did work on the system. Next I bring a hot object in contact with each black box. In each case the hot object is at a higher temperature than the black box. In each case, some energy spontaneously flows from the hot object into the system. I say I have caused heat to flow into the system. In either case, the energy of the system, the black box and its contents increases. To exclude the possibility that someone thinks that some of the energy of the system might have to do with the translational motion of the center of mass of the system, I tend to say that the internal energy of the system increases, rather than just saying that the energy of the system increases. I don't think I should have to include the adjective "internal" but it helps clarify things.
Up until a few years ago, I considered the internal energy to be the energy associated with unorganized molecular interactions and random molecular motion. Then John Denker brought up a situation along the lines of the one I am in the process of describing here. I open up the boxes. I find that one of them contains a gas. When I pushed down on the lid of the black box I compressed the gas. When I brought the hot object in contact with the black box heat flowed into the gas. But in the other box I find a bunch of gadgets and a cold potato. When I pushed down on the lid I pushed a gear rack down which caused a flywheel to spin. When I brought the hot object into contact with the black box a heat engine inside the black box caused the flywheel to spin faster and warmed up the potato. John's discussion of a similar situation convinced me that the internal energy of a system is the energy of the system whether it has to do with random interactions and motion at the atomic/molecular level or not. I started using the expression thermal energy for what I had been calling internal energy. The internal energy of a system became the energy of the system in the center-of-mass reference frame of the system.
Despite the many different ways that heat is defined, I have the impression that physicists have pretty much reached a consensus on a formal definition of heat as energy in transit, in particular the energy that spontaneously flows from an object to a colder object when the two are brought into thermal contact. Once it gets there it is not heat, nothing can contain heat, once it gets there it is part of the colder object's internal energy. ( I understand that John's position is that there is no such consensus. I think I have read all your arguments over the last few years about it, John; I still disagree.) So talking about the "heat content" of the ocean strikes me as wrong. It never occurred to me that a physicist might think that once that energy gets into the cold object it can be considered separately from energy that might enter the object by mechanical means and that the energy that got into the object because the object came into contact with a hotter object might be called the heat or thermal energy of the object in contrast with the energy that came in by mechanical means, and, it never occurred to me that a physicist might call the latter the work of the object. I thought John had constructed a straw man and was doing a good job of bashing it. Now I'm not so sure. I perused the Wikipedia entry on thermal energy and it is horrid. I'm still not convinced that most physicists who use the expression thermal energy think of it as the energy contained by an object that flowed into an object because of a temperature difference--perhaps none of the contributors to the Wikipedia article are physicists--but like I say, I'm not so sure. I am inclined to try to stop using the expression "thermal energy" because people might think I mean "heat content".
As regards the ocean, it has some energy associated with intermolecular interactions and random molecular motion and some energy associated with the bulk motion of water. They are both part of the total energy of the ocean. I think they are both part of the internal energy of the ocean. I think it would be very wrong to think that all the random molecular energy entered as a result of energy transfer owing to a temperature difference and all the energy associated with the bulk motion of the water entered by mechanical means. And yet, up until recently, I have been comfortable with calling one of them thermal energy and the other mechanical energy. There has, historically, been quite a bit of discussion on this list of thermalization. Stir some water in a bowl so that it has a lot of macroscopic angular momentum and wait a while. Wait until the water has stopped spinning. The water is said to have been thermalized. To me the verb implies that mechanical energy has been converted into thermal energy.
My current inclination is to go back to internal energy for the energy associated with unorganized molecular interactions and motion, use energy in the center of mass frame to include that and the energy due to bulk motion and arrangement.
The language is not precise. The water swirling around in the bowl helps to highlight that. There is a gray area between when the energy is kinetic energy of bulk motion and when it is energy of unorganized molecular interactions and motion. There is a gray area between when the things that change rapidly are done changing and when they are still changing. I don't think that the fact that boundaries are indistinct is a reason to stop using the terminology to talk about the things on either side of that boundary. John Denker has argued that as the difference between infrared radiation and microwaves is indistinct, the difference between heat and work as ways in which energy is transferred is indistinct. I agree but I don't think that that is a reason to stop using the words. In analyzing a turbine, it is useful to distinguish energy flow out of the turbine that occurs spontaneously because of a temperature difference and energy that flows out through the shaft mechanically. The words heat and work make communication easier.