A lot of the definitions of satellite states in political science literature tend to describe the Warsaw Pact states during Cold War period as well as Mongolia, so the definitions tend to fit these cases better than the Napoleonic period. Ferenc A. Vali wrote in his 1961 study of Hungary, Rift and Revolt in Hungary, a workable definition:

Political satellitism is a method whereby another country may be controlled or made dependent on another state without incorporation or any other legal or official admission of the de facto reality of dependency.

Owen Lattimore's study of Mongolia under Soviet influence also identified three preconditions for a satellite: a vested interest of the superior partner in controlling a state, a minority faction or clique seeking or enabling this relationship, and an overall fear that the alternatives to satellite status are less desirable than this form of limited sovereignty. Both theorists noted that satellite state rulers had a limited degree of autonomy, especially in places where the controlling party had little interest, but often such autonomy granted to the satellite states were to fulfill the needs of the dominant party.

Such definitions worked reasonably well for Cold War Eastern Europe which had their own national Communist Parties, armed forces, diplomatic recognition (save the GDR which was complicated), and other signs of independence, but were inextricably linked to Soviet power. The Warsaw Pact states governed themselves quite differently than a puppet state like Manchukuo, which had only the barest of pretenses to Manchurian sovereignty.

Things get a little bit dicey though with Napoleon's empire and its various dependencies. The Rheinbund states fit a number of Lattimore and Vali's major points quite well; states like Berg or Westphalia were nominally independent but Napoleon demanded both blood and treasure from them as well conforming to the Continental System. Joseph in Spain styled himself as a the new king in Spain even if his younger brother often micromanaged Madrid. The Duchy of Warsaw also fit into a number of the criterion; it was dependent on French power and its elite sought to ingratiate themselves to French power. However, the expansion of the French departments and various implemented or planned annexations of non-French territory do not quite fit the Cold War-era definitions very well. One of the basic rationales for a dominant power to up a satellite state was that outright annexation was more trouble than it would be worth or otherwise diplomatically unwise. Much of Napoleon's instincts were against brooking even a nominal degree of independence among his allies. When problems emerged such as in the Kingdom of Holland he simply annexed the Kingdom rather than lend aid to let his younger brother sort things out.

One other area shows a departure in the (admittedly under-defined) models for satellite states: defection. Outside of the Bonaparte clan, the rulers of the Napoleonic satellite states did have a degree of latitude over their own armed forces and diplomacy. While they feared French power, these rulers did recognize certain realities about Napoleonic rule. When French power was strong, they hewed towards Napoleon, but tacked away from Napoleon when his fortunes appeared on the wane. States like Bavaria or Württemberg had enough independence to time a defection where they could guarantee territorial sovereignty. Even the various Polish elites of the Duchy of Warsaw and Murat looked for a political lifeboat in the sinking Napoleonic empire in 1813. This does not quite fit Lattimore's notions of an minority elite rulers that have some sort of ideological or geopolitical connection with the dominant power.

Nonetheless, using twentieth-century definitions of a satellite state in the Napoleonic period is not like putting a round peg into a square hole. A more apt analogy is that it is more akin to trying to fit an oval into a circle; there will be significant overlap, but some key differences. Historians of the Napoleonic period generally do use the term satellite state/kingdom because there really is no alternative model that better explains this asymmetrical relationship.

I'm not aware of any hard and fast definitions, but I think you pretty much hit the nail on the head. Obviously the kingdoms he set up with his family members were satellites of the French Empire, but I would say that states which were also created and existed due to his direct patronage, such as the Grand Duchy of Warsaw, the Helvetic Republic, the Confederation of the Rhine etc. I'd say these would count, as after it was over they were all wiped off the map.

Prussia I wouldn't say qualified at all, as they were only militarily defeated and had lots of territory annexed by Napoleon, so they would be at best submissive but hostile.

Austria is also difficult because it was ruled by Napoleon's father-in-law, but it was still an entity which had suffered large scale humiliations and annexations by him, so I'd say they were just willing to bide there time and wait it out.

I'm not sure there is really a set criteria, it would have to be taken on a case by case.