Iodine is an element. There is no such thing as "synthetic" iodine. One iodine atom is just like another (different isotopes aside). The human body cannot determine the source of an iodine atom.
Seawater IS rich in iodine ions, so salt made from seawater would likewise be rich in iodine. The rest of your contentions may be true. I don't know.
Iodine in salt available at grocery stores, restaurants and in practically all processed foods, have synthetic chemicals added to them. These chemicals may include manufactured forms of iodide, sodium solo-co-aluminate, fluoride sodium bicarbonate, toxic amounts of potassium iodide, anti-caking agents and aluminium derivatives.
So your contention is probably correct, and the inference of synthetic iodine in articles most likely refers to additional synthetic chemicals added to manufactured iodide. Much of what we look for on the Internet when it comes to truth is like following a ping-pong ball; you can't always know where the ball will end up, or what the correct answer is. All you can do is make an informed decision based on your best guess!
Here is a pretty good article on iodine. It surprises me that the medical profession is still trying to figure out what the benefits and proper use is of iodine, doctors have many opposing viewpoints. The article does have a definition of synthetic organic iodine compounds, so synthetic iodine is a valid term.
http://www.westonaprice.org/modern-diseases/the-great-iodine-debate/
Forms of Iodine
IODINE (I2): Elemental iodine is available in a formulation called Thyactin by TriMedica, described as a “stabilized colloidal iodine preparation.”
IODIDE (I-): Elemental iodine is unstable so it usually combines with another element, such as potassium or sodium. Salt is iodized using potassium or sodium iodide. Potassium iodide (KI) is available in tablet form in doses ranging from 0.23 to 130 mg. Lugol’s solution contains 6.3 mg of molecular iodine/iodide per drop; Iodoral tablets contain 12.5 mg iodine/iodide each. Both Lugol’s solution and Iodoral are one-third molecular iodine (5%) and two-thirds potassium iodide (10%). Most formulations of tincture of iodine are a combination of iodine and sodium iodide. Supersaturated potassium iodide (SSKI) contains 19–50 mg of iodide per drop. SSKI tablets are recommended in case of nuclear accident, to protect the thyroid gland from radioactive iodine, but otherwise should be avoided.
IODATE: Iodine in combination with oxygen, such as potassium iodate (KIO3), is considered inferior to potassium iodide in terms of protection against radioactive iodine.
ENDOGENOUS ORGANIC IODINE COMPOUNDS: In food and in the body, iodine is usually bound with protein compounds. The main iodine-containing compounds in the body are the thyroid hormones thyroxine (T4, four iodine atoms joined to tyrosine) and triiodothyronine (T3, three iodine atoms joined to tyrosine).
SYNTHETIC ORGANIC IODINE COMPOUNDS: Drugs such as Amiodarone (an antiarrhythmic medication) contain iodine. The simplest organoiodine compound is iodomethane, used as a soil fumigant. More complex iodate compounds include nonylphenoxypolyethoxyethanol-iodine (C17H28I2O2) or Byacin, used as a germicide, as in teat washes.
DETOXIFIED IODINE: Sold as Atomidine, the manufacturing method is called a “modified detoxification process” which involves a stage in which electricity is run through the iodine in saline solution to produce a solution containing free iodine (see sidebar on Atomidine, page 43).
NASCENT IODINE: Similar to Atomidine, although requiring more electricity and a longer time to produce. The diatomic bond of the iodine molecule is broken and retains a high amount of electromagnetic energy. According to the manufacturer, “once in contact with fluids of the body, the charged atom of iodine starts a process of relaxation where it gradually loses energy over two to three hours.“