Roaming has become a popular addition to the metagame strategy, wherein one player does not lane nor jungle, but rather simply moves through the jungle/river to initiate ganks on lanes from level 1 and onward, until the teamfight portion of the game begins.
A roamer essentially guarantees that you're either running a solo lane, or two solo lanes if you have a roamer and a jungler. I've also seen a double roamer game played, though this tends to be less effective as enemies who are wise start playing defensively and reduce the effectiveness of your roamers, which put them behind in both CS and levels.
The qualities of a roamer can vary, but the general requirements are that they are mobile, can do burst damage, and have a strong stun or position altering (push, throw, etc.) ability. Taric, Evelynn and Alistar are all very popular choices for roaming. I've also seen Sion and Blitz play roaming effectively, but they were both out of a lack of a jungle spot on the team and normally suit better in a lane.
This strategy is similar to the "double jungle" mindset in that you get three solo lanes, with the exception that there is extra pressure on getting early kills to succeed. A definite risk vs. reward tossup, as successful roamers will quickly catapult your laners ahead in level / CS as well as demoralize your opponents, but unsuccessful roamers will fall just as far behind due to the lack of their presence in the laning phase.
You have one false assumption in your question: what you observed was not two quests, but rather only one.
In other words, the quest was "Capture Point X, while defending point Y from Capture". The quest lasts until one team manages to capture the other, which means, in your case, the 'defend' portion of the quest lasts until your team manages to capture the enemy point in question, rather than a simpler, "defend for X".
You could theoretically have to defend that point for the rest of the game, as long as neither of the two target points changes hands.
Best Answer
It seems to be bugged - I can confirm that ARURF does not count towards the completion of this quest. Both myself and a friend of mine have both had the exact same problem, where we win a game of ARURF which satisfies the requirements yet we don't get the unlock.
All other matchmade Summoner's Rift game modes should count, however. This includes Co-Op Vs. AI games, which is a quick and guaranteed way of finishing the quest.