There is no hard fast rule, only those agreed on by the owners; however, you would be hard pressed to find many keeper leagues that snake draft. I don't think I would join one under normal circumstances.
There might be some cases when it would appropriate. If the number of keepers relative to the total number of players on each roster is low would be such a case. Using your league as an example, if you only had 3 or less keepers, you might then have a snake draft as you basically are re-drafting your roster every year.
You mention that everyone keeps 6 players, but you don't slot them and don't have salary caps. I might suggest that you reconsider this, particularly if I don't ever have to drop a player. All it takes is one owner to end up with Trout, Cabrera, Scherzer, ....etc for his 6 keepers and he just holds on for their entire careers, while the guy who came last place has to keep 6 middle of the road players in his first 6 draft positions. Anyway, that wasn't the question you asked, I just saw your comment and thought I would put in my 2 cents.
Back to your original questions, in your league a straight draft is normal with 6 of 23 being kept.