The fact is, as in many other things, the terms being argued about have simply been used inconsistently; by users, by printers, and even by software vendors. So it's incumbent upon the designer to have at least an upper-level concept of the physical processes involved in order to discern the actual meaning from the specific context. But it's not rocket science.
Pixels Per Inch should be self-explanatory. A raster image is basically a string of color values arranged in a rectangular array. Each pixel (picture element) is just a data placeholder for a color value, usually (but not necessarily) rendered as a square. So it has no inherent size. (Runaway confusion has resulted from software vendors treating "Pixels" as if it is an actual unit of linear measure in the rulers of object-based graphics programs.) A pixel is actually infinitely scalable. So you have to specify the overall size of the image as a scaling factor. That's the "per inch" (or other linear unit of measure) part. It's a scaling value, and it is required to say anything meaningful about the measurable "size" of a raster image.
Dots Per Inch is far and away the most ambiguous term, having for so long been applied (and misapplied) to just about everything, because it's often the only term a speaker is even familiar with. It has for decades been used (and misused) interchangeably with, and instead of, PPI. Frankly, it should be abandoned, especially when the meaning is important, as in stating printing requirements. (Even though those stated requirements are often not really optimal.)
Lines Per Inch. Graphics, of course, quite often involve regions which vary in color. Litho printing devices normally don't print anything by physically laying down "smears" or "blends" of solid ink. The press simply either prints ink or prints no ink. So continuous-tone graduations and uniform tints (both of which can be required by either raster or vector elements) have to be simulated by printing an array of small dots of solid ink. (again, each dot in a given place on the page is either ink or "no ink." There is no "partial ink.")
That process of conversion to dots of ink is historically called screening because, long before computers, the high-contrast film used to expose the printing plates was itself exposed while in vacuum contact with a film (called a halftone screen or a tint screen) through which the original image of the photo or artwork (which is either continuous-tone or line-art) was projected on the process camera in the litho darkroom.
(So much confusion among beginning digital designers could be cleared up by a solid understanding of the differences between contone and line art. Unfortunately, you seldom hear those terms nowadays.)
Although nowadays, with computers, screening can be accomplished either by varying the sizes of uniformly-spaced dots (AM screening, borrowing the term Amplitude Modulation from wave dynamics) or inversely by varying the spacing of uniformly-sized dots (FM screening, borrowing the term Frequency Modulation) halftone screening is still most common and is still specified in terms of LPI.
Spots Per Inch. Here's another conceptual key that clears up a lot of confusion, not just about the various acronyms, but about the matter of printer resolution: Whether you are talking about AM dots (halftone screening) or FM dots (stochastic screening) each of those tiny screen dots is built up from much tinier spots of actual ink (or toner or whatever). These are the "dots" which the printing device really prints. They are all the same size. They are, in other words, the smallest mark the printing device can make on the page, and all larger marks are built up from them. That's why SPI is the term that should be used when you are talking about the actual resolution of the printing device, not DPI because, as explained, DPI has been so irreversibly overused for all kinds of things for too many decades.
Levels of Gray. Because halftone dots are made up of printer spots, the count of different sizes of halftone dots that a given printing device can print is a function of how many printer spots are available to render the dot. That count is how many levels of gray the given device can render at a given halftone ruling. That's why you see (and beginners are distraught over) banding in low SPI devices, like 600 SPI laser printers, that won't be a problem on a 2540 SPI imagesetter. It's also why buyers are often misled by manufacturer's stated resolution of desktop devices, like inkjet printers. ("Egads! 1440 DPI! And just 300 PPI is 'perfect'! What a miracle! What a deal!" Those 1440 are printer spots, and the screening method is usually FM, not AM.
Each halftone dot is made up of printer spots (SPI). The square array of pixels of a raster image is scaled to size (PPI) and spread across the array of lines of halftone dots (LPI), and the color values of the scaled pixels are averaged across that region of the page to determine the size of each halftone dot for each ink color.
Contrary to some misconception, everything--be it a raster image or a vector-based path--ultimately becomes one page-size raster image. (That's why it's called a RIP; Raster Image Processor.) The difference between raster resolution dependence and vector resolution independence is not whether it will be rasterized, but when.
By definition, the rasterization of raster images already exists when the raster image is created, so its "grid" is effectively "overlaid" onto the halftone grid and the printer spot grid of the device. So sharpness is dependent upon how the pixels are scaled and how those various "overlaid" grids correspond. The rasterization of vector-based graphics is deferred until print time. So vector-based graphics have no raster resolution until they are printed. They print at the actual appropriate resolution of whatever printing device(s) to which they are sent.
JET
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