Forza Horizon

There are 11 Lamborghinis in Horizon, with 7 of those being on disc, meaning you will have to purchase either a single car or a car pack (for this I suggest the November Bondurant pack to also grab the Snake Charmer achievement).

On Disc:
2012 Aventador LP700-4
1988 Countach LP5000 QV
1997 Diablo SV
2011 Gallardo LP570-4 Superleggera
1967 Miura P400
2010 Murcielago LP 670-4 SV
2009 Reventon Roadster

DLC:
2012 AventadorJ (October Pack)
2012 Gallardo LP570-4 Spyder Performante (November Bondurant Pack)
2006 Miura Concept (Season Pass only)
2011 Sesto Elemento (VIP only, available with Collector's Edition only)

You can't use Car Club garage cars. Only cars in your personal garage.

You need to earn at least one medal in 8 different Lamborghini cars. Press up on the D-pad to see the requirements for the medals.

Beware, although some cars are free to download, they still need to be purchased in game with either in-game credits or real money (MS Points), but that is the case for every car.

Unicorn cars
I do not include "unicorn" cars as free as you need to earn them and they are extremely rare, so most gamers will not have access to them.

Barn discovery cars
Barn discovery cars are available in the main game, but you have to progress through the story and multiplayer, to have them appearing on your map. Then you have to find them in the barn (wrecked). You have to wait until they are fixed before you can use them. There are 9 Barn discovery cars.
* 1964 Aston Martin DB5
* 1958 Aston Martin DBR1
* 1981 BMW M1
* 1992 Bugatti EB110 SS
* 1957 Ferrari 250 Testa Rossa
* 1956 Jaguar D-Type
* 1954 Mercedes 300SL Gullwing Coupe
* 1971 Plymouth Cuda 426 HEMI
* 1965 Shelby Cobra Daytona Coupe

Loyalty cars
To get your loyalty cars you need at least a save file from either FM3 or FM4. Depending on your level, you get those cars:
* 2011 MINI Cooper S (Level 1)
* 2011 Citroen DS3 Racing (Level 5)
* 2012 Dodge Charger SRT8 (Level 10)
* 1969 Chevrolet Camaro SS (Level 15)
* 2008 BMW M3 E92 (Level 20)
* 2012 Jaguar XKR-S (Level 30)
* 2009 Gumpert Apollo S (Level 40)
* 2012 Lamborghini Aventador LP700-4 (Level 50)

Forza Rewards Program
https://rewards.forzamotorsport.net/
Depending on your tier, you get rewards monthly. Redeem them on the site and they will be deliverer in your game within 4 hours. The rewards can be money, cars, tokens,...

Available DLC Packs
* October Car Pack (1957 Chevrolet Bel Air - FREE)
* November Bondurant Car pack (2012 Jeep Wrangler Rubicon - FREE)
* Season pass (2006 Lamborghini Miura Concepts - FREE)
* December IGN Car Pack (1971 AMC Javelin AMX - FREE)
* Rally expansion pack NOT FREE - NO FREE CARS
* January Recaro Car Pack (1995 Ford Mustang Cobra R - FREE)
* Honda Challenge Car Pack - FREE pack, includes 3 free Honda Cars (2013 Civic Si Coupe - 1986 Honda Civic Si - 2013 Honda HDP Fit - FREE)
* February Jalopnik Car Pack (1966 Ford Country Squire - FREE)
* March Meguiar's Car Pack (1998 Nissan R390 - FREE)
* April TopGear Car Pack (2002 BMW M3 GTR - FREE)

Complete List
* 1967 Miura P400 (main game)
* 1988 Lamborghini Countach LP5000 QV (main game)
* 1997 Lamborghini Diablo SV (main game)
* 2006 Lamborghini Miura Concept (FREE DLC - however I was unable to buy the car without spending MSP)
* 2009 Lamborghini Reventón Roadster (main game)
* 2010 Lamborghini Murciélago LP 670-4 SV (main game)
* 2011 Lamborghini Sesto Elemento (DLC)
* 2011 Lamborghini Gallardo LP570-4 Superleggera (main game)
* 2012 Lamborghini Aventador LP700-4 (Loyalty car)
* 2012 Lamborghini Aventador J (DLC)
* 2012 Lamborghini Gallardo LP570-4 Spyder Performante (DLC)

Summary: 6 cars (main game) + 4 cars (non free DLC) + 1 Loyalty Car

I suggest you buy the November Bondurant Car Pack as you need to buy it anyway if you want to unlock the next achievement: