I am not a strut or shock expert, but I have replaced several back in the day.......
It may not be your strut making the noise. Usually when struts go bad, it isn't noise that gives it away, it is your vehicle bouncing over bumps more than normal. Most often than not, noise, when made by struts and shocks are made by the mounts, shock or strut mounts that may be loose. Additionally, if they strut is not tightened to specs and is slightly lose, it will have a little play that could make the noise you described. The only issue then is why does the noise go away when the car warms up, for that, I have no answer. Lastly, your $200 price is low. Struts are expensive, moreover, they should be replaced in pairs. I would think new struts to the front of your car (pair, installed) would be closer to $800+.
Bottom line, if your car is not bouncing like a yo yo, and does not drive noticeably different, it may be nothing more than a worn mount or play which can be fixed when you finally do replace your struts. So, game on!