The Lake Forest neighborhood is actually located in Lacey. Lacey, Olympia and Tumwater are three distinct cities on the I-5 corridor that have grown together geographically. It's difficult to know where exactly one begins and the other one ends. There are lots of homes within your given parameters in Lacey or in Olympia, for sale or for rent.
Best of luck!
The Lake though is a big plus.
Your friend was wise to recommend Olympia! :) Oxford loop is in a neighborhood called Lake Forest. It is just a short drive from the growing Hawks Prairie area. The neighborhood offers community access to Long Lake, one of the premier lakes in our area. There is also an elementary school at the front of the neighborhood by the same name. You would be likely to find a home with the characteristics you described there and likely in the $180-220k price range.
I'd be happy to get you more detailed info, just shoot me an email! Good luck with your move!