Hi all, I am a member of SF Fixed gear as well as NYC Fixed gear forum. I am not sure about the SF, but the NYC Fixed gear forum is accessible only by invite and it is hidden from search engine, so not many people can find much about it unless introduced by someone else to it.
It seems that there is a confusion in some terms.

Here is what I understand.

Fixie - A name for any bike which does not have deraileur (or does not appear to have gears). Typically used by someone who does not own one.

Road Frame - A frame from a road bike which has dropouts (the place where is the rear wheel bolted in) either horizontal with an opening coming forward (older frames) or circular (newer frame). Road frame has attachments for shifting cables, has holes for derailleur etc.

Single speed bike - a bike which has only one cog in the back but can coast (ie, one can ride forward without rotating the peddals). Those bikes need to have brakes (at least one - typically front one). Some single speed bikes have a 'back' brake which means that they have to attempt to pedal backwards to brake.

Track bike bike - a bike which consists of 'track frame'. The sure way how to recognize a track frame is to look at the rear dropouts. The dropouts are horizontal and the opening is pointed backwards. That is to make sure that if one pushes (pedals) really hard, the tension of the chain cannot cause the rear wheel to slip out of the drop-outs (as it can happen if the wheel was in the older road frame with horizontal dropouts but facing forward). Track bikes do not have a brakes. The true track bikes have undrilled forks (there is no hole to put a front brake in) and undrilled rear brake bridge (if there is any). Also the width of the rear wheel stays is smaller. The only way to stop track bike is to resist the constant rotation of pedals. One can either resits gradually and stop slowly or stand up and lock knees and essentially stop the pedals with the rear wheel in rotation and skid to the stop.

Many guys will ride a track bike on the street without or with the front brakes. Among very young ones, more street cred have bikes without the front brakes but general fixed gear community encourages having a front brake. For that one has to either drill the fork to be able to attach it or to buy a special "Keirin' brake which can be attached differently.

Most common are so called Conversion bikes - bikes which utilize old road frames with horizontal dropouts. They can be find cheap, made from amazing steel and can look very classy. They are already driled to accommodate both, front and rear brakes. One problem is that the track wheel (wheel with just one cog) is slightly narrower in width than the space between dropouts and the dropouts are facing forward, so the bikes might be not as safe when hard pedalling or skidding to a stop. The rear wheel might actually slide forward out of the dropouts due to the chain pull.

Conversion can be also a 'single speed bikes' - that means that they have one cog but can coast. Many wheels accommodate the option by having fixed gear cog on one side and coasting cog on the other.

That said, I am riding conversion with BOTH brakes and 'fixed gear' cog. It is still my favorite ride. I rode many centuries on ratio 48/16. Yes, it is quite a workout. Because of the rear wheel issues (rear wheel slipping out), I recently bought a true 'track frame' which I am going to be using instead of the conversion frame I had. I will still use BOTH brakes (I am too old to skid), the frame is drilled already for them.

Anyway - sorry for the long post. I thought you might find it useful.

Martina