Admin script still prints the error message for everyone if they don't use the prefix?

There is a small syntax error:-

-- you missed the  .. after player.Name
print("[LINC] "..player.Name " tried to use the 'reset command!'")

The second possible problem is the way you have setup the logic so:-

local Admins = {["mosski123"] = true, ["Player1"] = true}
local prefix = ';' 

function onChatted(message, player)
	-- we run this if statment
	if message == prefix.."reset" and Admins[player.Name] then
		-- code
	else
	   -- code
	end
	
	-- we also run this statment even if we have ran a command
	if message == prefix.."rejoin" and Admins[player.Name] then
		-- code
	end
end

-- we should use elseif as only one of the ifs will run as only one command will do one thing
if [condition] then
	-- code
elseif [second condition] then
	-- code
end

There are some other things that should be changes in your code, firstly I would recommend that you use UserId instead of the players name as the id will not change. Secondly you do a lot of processing that can be done once ie, prefix.."reset" and Admins[player.Name].

Example:-

-- we should use the player id not their name
local Admins = {["1540993"] = true, ["0"] = true} -- my id ect
local prefix = ';' 

-- just an idea to remove the long if statements, you can still use your method but you would need to use elseif
local commandTable = {}

function commandTable.rejoin(message, player)
	player.Character:BreakJoints()
end


game.Players.PlayerAdded:connect(function(player)
	if Admins[tostring(player.UserId)] then -- we only connect the admin command chatted event for admins instead of checking this each time
		player.Chatted:connect(function(message) 
			if message:sub(1,1) == prefix then -- we check for a prefix before checking for commands
				
				-- we remove the prefix its not needed
				message = message:sub(2, #message)

				if commandTable[message] then -- we have a valid command
					commandTable[message](message, player) -- we run the function in the table passing it the same data as in your code.
				end
			end
		end)
	end
end)

I hope this helps, please comment if you do not understand how / why this code works.