I would also tend to agree with both descriptions above. However, the exaggerations involved are often not so obvious as simple frequency response variations. A speaker can have a relatively flat on-axis response yet still sound "hyped".
Here are some examples. Let's assume they all measure flat on-axis:
Larger 8" and 10" 2-way speakers can sound scooped in the midrange because their woofers become very directional in their upper frequency range, beaming like a flashlight. Since there is less off-axis energy (usually in the range of 1kHz to 3kHz) the speaker will sound recessed in this range, coloring the mix with a pleasing "smiley face" EQ character.
Smaller 5" and 6" 2-way speakers can sound punchy in the upper bass and lower midrange. When smaller woofers are pushed to give deeper bass response than they can optimally reproduce they also generate quite a bit of harmonic distortion to go with it. These harmonics land right in the upper bass and lower midrange giving a sense of punch that doesn't actually exist in the mix.
Ported and passive radiator speakers will sometimes have "robust" sounding bass. The ports and passive radiators are basically resonators that ring out and enhance the bass output. These resonators sacrifice quick transient response but their slower blooming character can make certain bass notes sound like they have more authority.
Yes, my designs are 3-way sealed box. So I do have some prejudices.