Like much JEE/NEET preparation material, the text states somewhat idiosyncratic definitions as universal fact and isn't quite right (and isn't proofread—see "liqufied").
I believe the intended distinction the text aims to make is to define "vapor" as the state of matter with no surface tension and temperature lower than the critical temperature. (The critical temperature is the temperature where the surface tension of the liquid phase drops to zero.) This corresponds to the tan region below, using the phase diagram of water as an example.
Observe that condensation into the liquid phase—the green region—becomes spontaneous if the vapor pressure is increased to the level of the black separation line.
In contrast, "gas" appears to be defined in the text as the state of matter with no surface tension and temperature above the critical temperature, corresponding to the blended region to the right of the green and tan regions. Here, no line of demarcation can be drawn between the liquid and gas phases because the properties change smoothly (from 0 Pa to about 10 GPa above 647 K, for example). In other words, "gas" can't be condensed into a liquid in well-defined phase transition simply by applying pressure; for that to occur, the gas must first be cooled below the critical point, at which point it's called "vapor," again according to the text.
To repeat, these are somewhat idiosyncratic definitions. Others would define the entire region of no positive surface tension as a gas (subcritical or supercritical, with a supercritical liquid blending into a supercritical gas in terms of material properties), and distinguish vapor specifically as gas in equilibrium with the condensed phase. (So humidity in air would correspond to gaseous water, or alternatively water vapor that at one time evaporated from liquid water or ice.) To imply—as the text does—that vapor isn't gas would elicit quizzical looks from most physicists, I think.
The text is also incorrect that the vapor pressure is independent of the external pressure. The dependence is small but not zero.