How do I detect a player has spawned after a death rather than entering first time?

Died

Humanoids have an event called Died. This event is fired when the player dies. You can use access the Humanoid by going through a player's character.

You can use this to perform actions when the certain players. For instance:

game.Players.PlayerAdded:Connect(function(plr) -- fires when a player joins
    plr.CharacterAdded:Connect(function(char) -- fires when the character loads
        char.Humanoid.Died:Connect(function() -- fires when the player dies
            print(plr.Name .. " has died!")
        end)
    end)
end)

Wikia Links

Died Event

I'm assuming that you want to do this on the server.

You can use a table with Players as keys and booleans as values; when a Player first joins, assign their Player Instance to true. Whenever their character is added, check if the boolean is true. If so, assign it to false. If not, continue as normal to fire your event, execute your code, etc. We'll also remove the Player's field when they leave, to prevent a memory leak:

-- Services:
local players = game:GetService("Players")

-- Variables:
local player_first_chars = {}


-- Main:
players.PlayerAdded:Connect(function(player)
    player_first_chars[player] = true
    player.CharacterAdded:Connect(function(character)
        if player_first_chars[player] then
            player_first_chars[player] = false
        end
        -- ...
    end)
end)

players.PlayerRemoving:Connect(function(player)
    player_first_chars[player] = nil
end)