Registered: Sep 2005
Here's my take on it.
Recall the republic existed up until episode 3. Palpatine already being the leader of the republic, transforms the current government into the first galactic empire. The empire at this time is basically the only government in the galaxy. A rebellion then forms to fight against the empire to restore peace in the galaxy. Rebellion is what you would call it as the empire controls the galaxy in episode 4-6.
Now since the republic is restored, and by episode 7, the new order is basically like an opposition party in the current government. The republic and first order both exist at the same time and are obviously in disagreement. General Hux basically talks about this right before the star killer base destroys the multiple planets. Part of the republic supported planets get destroyed. Recall Finn saying "it was the republic, the first order did it". Hence the first Order, led by supreme leader snoke, is taking back the galaxy by force or basically destroying the republic.
Now, the resistance is like an independent group working to directly oppose the first order only, since both the republic and first order both exist. First order kinda like being separatists but more evil.
Perhaps, the republic directed there military funds to fund the resistance rather than built up on a new super power army. Recall in episode 6, the rebel alliance didn't seem like they had more of a military force, then the empire. Also, in episode 2, it took quite a bit for the republic to approve a grand army to fight against the seperatists. They also had more Jedi in episode 2
Also, before episode 7, Luke did try to train new Jedi in an academy as Han Solo says, but doesn't actually give a number. Only that Ren turned to the darkside. So, perhaps, the republic was partially counting on this new Jedi order that Luke apparently failed at.
So, I can see that a resistance military force is the proper term to be against the first order.