It's not the exaggeration that is the problem. The article uses two completely different terms interchangeably what is an example of ignorance on the writer's part. The title says "US has been 'at war' 91% times of its existence' and then follows with 'United States has been involved in some war or the other in 214 out of 235 years [...]'. Being 'at war' is something completely different from 'being involved in war'. The former is the very precise legal term, the latter is very vague expression. Describing red squares as years US was 'in conflict' obfuscates the issue even more as conflicts can be also of non-military kind (blockades, area contest, unrest etc.). Being 'at war' or 'being in the state of war' means the fact of having declared war on another country or having the war declared on oneself by a sovereign, recognised state (the adjectives are important, as say, suppressing a secession does need to qualify as a war in terms of international law). Being involved in a war may mean a lot of different things, for example, being invaded without declaration of war and with no retroactive declaration of war by the invaded state (case on Soviet invasion of Poland in 1939), a military support for the country involved in a war, an economical or even political support for any country in the state of war etc. It is worth noting that neutral states can still be 'involved in war' by, say, selling weapons or other supplies to one or all involved sides or granting asylum to war refugees.

So, summing it up, the correct way to present the involvement of US in wars is to say that the Unites States was involved in wars for over 200 years out of 235 (or 242 in 2018) but was at war only for less than 11 years in total, 16 years if count in the American Civil War or 24 years if also count the American War of Independence what is stretching the definition of being 'at war'.

So: the image in question says that its source is "Loonwatch.com" - which doesn't strike me as a very credible source (it seems to be an anti-Islamophobe blogzine, and maybe they mean well, but I wouldn't rely on them for my history). The link to the other question sides a different blog, which at least provides some additional detail as to what events it is talking about in what years.

Now, that second list certainly looks familiar to me, and I strongly suspect it is based off of this report compiled by Richard F. Grimmett, a National Defense specialist, for the US Congressional Research Service, titled "Instances of Use of United States Armed Forces Abroad, 1798 - 2004".

I feel compelled to quote the introduction, so bear with me:

"This report lists hundreds of instances in which the United States has used its armed forces abroad in situations of military conflict or potential conflict or for other than normal peacetime purposes. It was compiled in part from various older lists and is intended primarily to provide a rough survey of past US military ventures abroad, without reference to the magnitude of the given instance noted. The listing often contains references, especially from 1980 forward, to continuing military deployments especially US military participation in multinational operations associated with NATO or the United Nations. Most of these post-1980 instances are summaries based on Presidential reports to Congress related to the War Powers Resolution. A comprehensive commentary regarding any of the instances listed is not undertaken here.

The instances differ greatly in number of forces, purpose, extent of hostilities, and legal authorization. Eleven times in its history the US has formally declared war against foreign nations. These eleven US war declarations encompassed five separate wars: the war with Great Britain declared in 1812, the war with Mexico declared in 1846, the War with Spain declared in 1898, the First World War, during which the US declared war with Germany and with Austria-Hungary during 1917, World War II, during which the US declared war against Japan, Germany, and Italy in 1941, and against Bulgaria, Hungary, and Rumania in 1942. Some of the instances were extended military engagements that might be considered undeclared wars. These include the Undeclared Naval War with France from 1798 to 1800; the First Barbary War from 1801 to 1805; the Second Barbary War of 1815; the Korean War of 1950-53; the Vietnam War from 1964 to 1973; the Persian Gulf War of 1991, global actions against foreign terrorists after the September 11, 2001 attacks on the United States, and the War with Iraq in 2003. With the exception of the Korean War, all of these conflicts received Congressional authorization in some form short of a formal declaration of war. Other, more recent instances often involve deployment of US military forces as part of a multinational operation associated with NATO or the United Nations. Current legislation, H.R. 10 and S. 2845, contains provisions that could strengthen the intelligence capabilities of the United States to combat international terrorism and to support current and future American military operations overseas in a more effective manner.

The majority of the instances listed prior to World War II were brief Marine or Navy actions to protect US citizens or promote US interests. A number were actions against pirates or bandits. Covert actions, disaster relief, and routine alliance stationing and training exercises are not included here, nor are the Civil and Revolutionary Wars and the continual use of US military units in the exploration, settlement, and pacification of the Western part of the United States. This report will be updated as warranted.

So this is a compilation of any instance that any members of the US military were deployed beyond then US borders. To Washingtonsblog's credit it looks like they added various Indian Wars to that list. To its detraction, it takes a lot of deployments to protect US citizens' property or embassies, or to pursue pirates, and turns them into "wars."

So you get on the blog post:

"1822 – Anti-piracy war (see note above)"

While the CRS report states:

"1822 -- Cuba. United States naval forces suppressing piracy landed on the northwest coast of Cuba and burned a pirate station"

Which seems pretty weak sauce for a "war".

Which gets to a the larger question: what counts as a "war"? The CRS is using a maximally-expansive definition in order to cover all US military deployments abroad, while noting that there are a much more limited number of formally-declared military hostilities, with a number of undeclared conflicts (Korean War and Vietnam War, for example) as runners up, if you will. It does not include such domestic conflicts as the Civil War, or Indian Wars, or even the Whiskey Rebellion. It also doesn't include conflicts between substate or non-state actors, so no Detroit Riots or Range Wars.

It may be surprising, but there is no universal definition of "war". Political scientists engaging in conflict studies will often try to come up with definitions in order to create a baseline for analysis. Here is one from the Center for Systemic Peace (a conflict studies nonprofit in Washington, DC):

""Major episodes of political violence" involve at least 500 "directly-related" fatalities and reach a level of intensity in which political violence is both systematic and sustained (a base rate of 100 "directly-related deaths per annum"). Episodes may be of any general type: inter-state, intra-state, or communal; they include all episodes of international, civil, ethnic, communal, and genocidal violence and warfare. "

They used that definition to compile their "Major Episodes of Political Violence 1946-2016" list, which tallies 328 armed conflicts across the globe since 1946, with 36 ongoing as of 2016.

Other definitions used by academics or organizations may be different, but will be roughly similar: there must be at least two identifiable "sides", x number of directly relatable deaths (1,000 is often picked) and at least y number of deaths per year.

But these are always definitions used for classification in hindsight: no one plans to kill at least 1,000 people through direct actions with at least 100 killed per year. Sometimes armed groups are mobilized and no one is killed (or even no casualties occur), and a small firefight between a few people which doesn't kill your cutoff can certainly feel like a war.

So - there is no universal definition of war. And as far as I can tell no one has attempted to catalogue every possible act of politically-motivated violence that has involved the US government or American citizens. I'm sure that would be a quite long list if done, but I am equally sure that it would a long list for almost any polity, at least one that aspired to some sort of military or even economic power status.