Everything else using the term "level" in Dungeons and Dragons makes sense. A third level character has three hit dice. A third level monster is typically found on the third level of a Dungeon. The third level of the dungeon is found beneath levels one and two.
So why does a third level wizard get one second level spell?
This really should not be that hard to fix. Simply take the first level spells, separate them, fairly evenly, between weaker and stronger spells (perhaps putting the combat spells in the stronger spell category) and re-label the stronger first level spells "level two spells". A first level magic user selects a spell from the weaker 1st level spell list. When the magic user reaches 2nd level, she can take a spell from either the first or second level spell lists.
Same with former second level spells. Separate them, and re-label the weaker as third level and stronger as fourth level spells. The 3rd level magic user takes a spell from the 3rd level spell list, and when she makes it to fourth level, takes one from the 1/2 list, and one from the 3/4 list.
The same applies to third level spells, which become fifth and sixth level spells, and so on, until we reach the ninth level spells, which are separated and become levels 17 and 18, respectively.