They are wrong, for the most part.
The heart is contains what are called "autorhythmic" cells. These are muscle cells that do not require stimulation from the nervous system.
This is why a heart completely removed from the body will continue to beat. All the central nervous system does is fine tune the beating.
The baseline rate of the heart is dictated by the fastest autorhythmic cells, which are called the pacemakers. In normal physiological systems the pacemakers are the SA node.
The SA node is in the right atrium. It triggers the right atrium to beat which pumps blood into the right ventricle. The signal then travels to the AV node and through the Purkyně fibres down to the base of the heart. From the base of the heart the signal cascades through up through the muscle cells in the ventricles which pumps blood to the lungs (right ventricle) and the body (left ventricle).
The reason the signal travels to the bottom of the heart and then the heart muscle constrict upwards is that all of the blood leaves the heart through the top. So, the heart muscle fibers actually twist like a helix, and constrict form the bottom to allow as much blood to be pumped out as possible.
The way signals in the heart travel from the SA node through this complex system is what are called Gap Junctions between the cells. These gap junctions are literally pores on the membranes of the cell which are surrounded by proteins which keep the pore of one cell closely bound to the pores of adjacent cells. The locations of these pores are highly regulated (you don't regrow heart cells).
Now, the way a striated muscle (skeletal muscles and cardiac muscles, as opposed to smooth muscles) contracts is sort of complicated. There are two protein chains in close proximity to each other. One is like a bunch of little hands sticking, and the other is like a rope with handholds. The rope is anchored to a giant protein disc, which is anchored to a "tansmembrane" protein, which is then anchored to the connective tissue surrounding the cell. To regulate when the hands can grab the rope, the rope is covered in another protein which blocks the handholds. To move this protein calcium has to bind to it, and when calcium binds to it it moves out of the way and allows the hands to grab the handholds. Then the hands pivot using energy from ATP bonds to pull the rope which shortens the muscle and causes contraction.
The issue is that if calcium is allowed to just float willy-nilly around the cell it will precipitate out solution and crystallize (this is how bones are formed, and unregulated it causes a disease where the soft body tissues calcify). To prevent your muscles from becoming fossils inside of you, the calcium is stored in specials sacks, and are bound to another chemical.
Before I go on you need to know that there are three types of "gated "channels in the membrane of cells - voltage gated, ligand gated, and mechanically gated. They are gated, because some mechanism is preventing whatever is supposed to pass through the channel from being able to. A voltage gated channel has a protein which changes shape in depending on the electric potential on either side of it. A ligand gated channel has a protein which changes shape when a particular chemical attached to (and that specificity comes from the shape of the protein). A mechanically gated channel is exactly what it sounds like, when the membrane changes shape due to some force the protein also changes shape.
So when a muscle wants to contract they have to release the calcium from that sack. In skeletal muscles a nerve sends a signal down its axon in the form of an electrical gradient created by a concentration of sodium ions near the membrane of the cell. When enough sodium enters a local area it reaches a threshold that opens voltage gated channels. Those voltage gated channels allow even more sodium into the cell opening more voltage gated channels. The reason the voltage gated channels open is because sodium ions have a charge, so a gradient of sodium ions is also an electrical gradient (which is what voltage is - electric potential). When that signal reaches the synapse (where the nerve almost touches the target cell) it opens specials channels in the membrane which allow a few calcium ions into the nerve cell. Those calcium ions then trigger a cascade of responses which culminate in acetylcholine being released into the synaptic cleft (the gap at the synapse). Acetylcholine then binds to a ligand gated channel in the muscle cell which allows sodium to enter the muscle cell. If enough sodium enters the cell a voltage threshold is reached and another signal is sent along the surface of the cell until it reaches the special sacks in the muscle cells containing the calcium. Once there a mechanical gated channel opens in the special calcium sack which releases a huge flood of calcium into the cell.
Heart cells are more complicated a bit more complicated.
What is most important is that autorythmic cells are kind of "leaky" to sodium ions, and so sodium is constantly leaking into the cell. Meanwhile, the cell is using ATP to try and pump sodium and potassium out of and into the cell to maintain a gradient (gradients are how all work is done in the entirety of the universe). Things want to move down a gradient. Heat dissipates into cold areas, water flows down hill, chemicals want to disperse evenly throughout a fluid. When enough sodium leaks into the cell the threshold is reached, and the cell "fires" which concludes in the heart cell contracting.
Want ends up happening is that instead of some stimulus creating the gradient, the autorythmic cells are so leaky to sodium that they are constantly being triggered with a regular time interval. If this occurred in a skeletal muscle you would have a severe twitch.
To conclude, the heart begins beating because some sodium leaked into the SA node creating what is called an "electrochemical" gradient.
If I were to take a stem cell, and force that stem cell to form into a cardiac muscle cell, and then put that muscle cell into a solution of saline (salt water), the cell would start beating.
But hey, don't take my word for it. Here is a video of it in action: https://www.youtube.com/watch?v=WLtJTywFpoI