Actual definitions vs media definitions, and translation in to English
As Glenn has said, most people don't care, and won't be changing their minds, even if one explains the difference in terms, and how to correctly use them.
Back in the tail end of WWII (1944) the Germans fielded a new class of firearm. It fired a round less powerful than their standard rifle, yet more powerful than a pistol. It fired in both semi-automatic and full automatic. When Adolf Hitler approved the weapon, he christened it the "Sturmgewehr".
Sturmgewehr translates into English as "assault rifle" or "storm rifle" The words assault and storm are mean in the military sense, that of assaulting, or storming, an objective. NOT in the civilian sense of one person assaulting another.
The media (and most people) don't care about the difference. They only see the English word assault, and assume it refers to one person assaulting another, as in the legal definition of criminal assault. They say "any gun used to assault someone is an assault rifle", or some such twaddle. Technically, in English, they are correct, BUT they have altered the meaning of the term completely.
The term "assault rifle" has been in use by the military and shooting community for over half a century, the guns being defined by their ability to fire selectively (both full auto and semi) at the flip of a switch, and the power level of the ammunition they used.
Back in the late 1980s, the anti gun people started a campaign to confuse the issue, willingly aided and abetted by the press. They began by calling semi auto guns that looked like military guns assault rifles. When shooters pointed out that they were using incorrect and improper terms, the started calling the same guns "semi automatic assault rifles". This proved to be too much of a mouthful to make a good sound bite, so the media eventually dropped the "semi automatic" part.
THEN, to further advance their agenda, they came up with the term "assault weapon", a term they completely made up, which included semi auto rifles, pistols, and shotguns with features they dis-approved of, such as pistol grips, detatchable magazines, flash suppressors, bayonst lugs, barrel shrouds, folding stocks, etc. By 1994, they had gotten Congress ( and several states) to actually make this defintion legal and put into law.
Under US law, actual assault rifles are legally machine guns, and have been regulated as such, since 1934. Since 1994, semi-auto guns with certain cosmetic features have been legally classed as "assault weapons".
And, since people tend to be sloppy and imprecise in their language, and since dictionaries (especially on line ones) reflect the definition of words as found in "popular usage", dictionary "definitions" of what is an assault rifle are seldom correct in the legal and technical sense. The field of firearms terminology, as defined in law, is arbitrary, confusing, often ambigious, and sometimes seems to contradict basic common sense. The terms used by the media are even more imprecisely applied.
As to what constitutes a "high power" cartridge, again, the media has a different definition than the shooting community. They tend to refer to everything larger than a .22 rimfire as a "high powered" round.
When the .30-30 winchester came out, over 100 years ago, it was considered a high velocity, high power round. It became, and still remains the most popular deer rifle round in the US. But by todays standards, it is not a "high power" round. The 7.62x39mm round used by the AK and SKS series of rifles is balistically very slightly below the energy of the .30-30 Winchester. It is not considered by shooters to be a high powered round.
The 5.56mm (.223 Remington) round used by the M16 and AR 15 series of rifles is also not considered a high powered round. It is a varmint cartridge, and not legal for deer hunting in the majority of the US, because it is not "powerful" enough to meet many state Game Depts requirements. But the US military considers it enough to shoot enemy soldiers with.
The media plays fast and loose with terminology, firearms, and otherwise, because they are in the business of selling news. Precise accurate reporting does not sell as much as sensational sound bites and catch phrases. And the majority of politicians and police officials quoted by the media don't have any better education in proper firearms terminology than the press does.
If you are looking for the truth, you will seldom find it in the media, the politicans, or Wickipedia.
Hope this helps answer your questions.