Unfortunately, "Motorist Overtaking" type collisions as defined in the bicycle crash literature and the most commonly used crash documenting software, PBCAT (Pedestrian and Bicycle Crash Analysis Tool), are much more frequent than that and result in a disproportionate portion of fatalities.

Some specifics copied from a thread on another bicycle forum:

The often cited major US study is Bicycle and Pedestrian Crash Types of the Early 1990's (FHWA-RD-95-163, a 1
. Of their 85 crash types, 5 (6 including "Type unknown) are in the "Class D: Motorist overtaking bicyclist". They compared their data to the 1977 Cross and Fisher study and for this class of crashes:

1977 study, Total Fatal=166 of which 37.8% were motorist overtaking, Total Nonfatal=753 of which 10.5% were overtaking
1990's study, Total Fatal=41 of which 29.3% were motorist overtaking, Total Nonfatal=2453 of which 9.8% were overtaking.

As you can see, the Motorist Overtaking class has a disproportionate fraction of the fatalities in both studies and and was not a rare crash.

However, all of that data is (obviously) before cell phones, texting, and a variety of other new causes of distracted driving as well as possible improvements in highway safety such as better motor vehicle headlights, better bicycle reflectors, better bicycle lights etc. and improvements in emergency trauma care. Some more recent data:

Wisconsin 2003
City crashes: 5.22% motorist overtaking;
but for Rural crashes 12.02%; Motorist Overtaking Undetected Bicyclist was the most common crash type.

City of Toronto Bicycle/Motor-Vehicle Collision Study, 2003 (city, not rural) found motorist overtaking was 11.9% of collisions, the second most common type, accounting for 4 of the 10 fatalities:

N. Carolina crashes from 2001 through 2007 rural/urban vs. crash type and aggregating the motorist overtaking collisions you get:
Urban, motorist overtaking: 373 of 4550 = 8.198% of all crashes
Rural, motorist overtaking: 553 of 2194 = 25.205% of all crashes

Further details on the N. Carolina data including graphs showing a steady increase in the percent due to motorist overtaking to about 30% of rural crashes in 2007 in this thread.

However, using that statistic for getting hit due to bicycle braking is a gross misuse of the data because in the large majority of the cases (see edit below) the bicyclist was not stopping and getting run over!

I think one of the reasons brake lights on bicycles may not be as important as on cars is because under good braking conditions (dry pavement etc.) four wheel motor vehicles can stop much faster than an upright diamond frame bike. Wilson's Bicycling Science 3rd edition goes through the physics. If your bike's brakes are good, you go over the handlebars at about 0.5g deceleration. Motor vehicles on dry pavement can brake to the limit of adhesion of the tires on the pavement, about 0.8 g deceleration. That means they can stop in 40% less distance than you can (0.8 / 0.5 = 1.6, so 0.4 less = 40%). Wilson adds this is why cyclists should never follow closely behind motor vehicles; they can stop a lot faster than you can. The exception (in principle) are tandems and recumbents. If their brakes are good enough, they too can brake to the adhesion limit (but with only two wheels and no anti-lock, maybe not quite as fast). Still, I wouldn't draft a tandem or recumbent without both of us being aware of the potential differences in braking ability.

The brake lights may very well help (more in an urban environment?), but probably a rear view mirror, some looks back over the shoulder, being highly visible from the rear, etc. are more likely to reduce your chance of being run over in general, not just while braking you bike.

Edit: Since depending on how a PBCAT database is set up, it may be ambiguous in the "Overtaking" categories cross tabulated with the "Fault" field if braking was involved. I went back to the Bicycle and Pedestrian Crash Types of the Early 1990's and on page 86 the table for Crash Characteristics lists under Bicycle Maneuver "Slowing Stopping 1.2%, Stopped 0.9%". Thus at most, with 10.8% of crashes in the Motorist Overtaking types, only (1.2% + 0.9%) / 10.8% = 22.7% were associated with bicyclist slowing or stopping. Old data, but I suspect it may not have increased that much and thus is at most a minor part of the "Motorist Overtaking" types.