Has anyone on here used an alternative to XP in their campaigns that they've participated in as a means to level up?
Yes I have, and more often than not actually.
Was this new way to level brought about by the players or the GM?
Both, I have been a GM more often than not. One game was only run on a monthly basis and the players suggested I level them up after major plot advancing sessions. Currently, I suggested a similar method due to having 9 players who play on a fluctuating basis and wanting to keep them all at the same power level.
And lastly, by having this new method to level up, did you see any change in behaviour from the players? (i.e. Problem solving to achieve their objectives through stealth, diplomacy, responses like that over killing whatever moves)
Yes and no. The group I had that met monthly was always more story driven than combat driven, so when we decided that leveling after plot points nothing changed. They had always worked toward advancing the plot and continued to do so.
My current group is more combat oriented however, and while they still really enjoy the combat, I feel like they are more accepting of any RP or non-combat encounters now that they know XP is a non issue. I haven't talked to them about it but I suspect that since they thought fighting was the only way to get lots of XP quickly, that was the reason they wanted to focus on it so much. (Even though I gave them XP for RP and and non combat encounters the players had it in their head they had to kill to level up).
Alternatives to XP Rewards
First, you can just level them when you feel it's necessary, with nothing other than your whims of power to guide you.
Second, The DMG offers a suggestion of granting levels every 6-10 encounters depending on the difficulty of the encounters. Harder encounters level characters more quickly than easy ones.
Third, You can grant levels after Major plot development, such as finding a special magic item, saving a damsel or completing any number of quests.