Wyosmith has it exactly right.
A rifle is easiest to hit with, hits with the most authority, and with just a little attention to ammo selection, less of a liability from a miss or over-penetration. (A common misconception that I see at least one poster in this thread believes. Reduced penetration in building materials from AR-15- power carbines, when compared to pistol-caliber subguns, is just one of several reasons most LE special response teams have gone to AR-15s or similar platforms.)
Additionally, modern intermediate power carbines are much easier to train up with than shotguns or handguns, and much easier to fire quick follow-up shots with acceptable accuracy.
Now, if a TFL member does most of their shooting with a shotgun, then a shotgun is probably the best HD choice for them, just because of training. If a shooter only has a shotgun, it can work fine, despite not being optimal.
Ultimately, the biggest difference is not between a carbine or rifle and some other gun. The big difference is between having a firearm and not having one. Train with what you have, make good plans (calls, contact info, fire extinguishers, insurance, lighting, transportation, water, food), and be happy.
If you have the time and free funds to get a good modern carbine, AR-15 carbines are at an all-time low price when adjusted for inflation. Paying your utility bill is probably more important, but now is a good time to buy one, if you can.
John Shirley