There is no such thing as a “game mode” in Minetest.
There are several settings which are more or less independent from each other.
There is Creative Mode which allows you to place stuff freely.
There is PvP which allows players to attack each other. If PvP disabled, players can NOT attack each other!
You can also toggle damage.
And much more.
But these settings apply to all players, there is no exception. But mods can mess around with Minetest a lot, so mods *might* be the reason for your problem, but I am not sure about that. But yeah, please post the mods you are using, just in case.
Anyways, you can combine these settings as you wish.
For “survival”, you probably want to disable Creative Mode, enable PvP combat and turn on damage.
PvP can only be edited in minetest.conf (I don't know the default setting), the other settings are changable in Minetest before you start the server.
For reference, here are the relevant settings for minetest.conf:
Creative Mode disabled: creative_mode = false
PvP enabled: enable_pvp = true
Damage enabled: enable_damage = true
You don't need to be admin of the server directly if you have set these values *before* the server was started, just edit the minetest.conf (or the in-game checkboxes) before you start the server. But if you want to change these settings in-game, you need the “server” privilege.
However, as far I know, some of these settings also only apply after a server restart, sorry. :-(
For more information about working with servers, look here:
http://wiki.minetest.net/Server