Can a problem with starter cause battery light to come on

About six months ago my battery light started flickering very rapidly suddenly. I stopped and had the battery, alternator and starter tested and the person said my starter was fine but wasn't able to test the alternator or battery for some reason. I light went off for two months and suddenly started flickering again for a little bit off and on for about a week then had to be jumped started to start it. I replaced the battery and no issues again until now, three months later from replacing the battery. The light started flickering again. So I just bought an alternator thinking it's that even though the light has stopped again. I haven't had the opportunity to put the alternator in yet but my starter just stopped working today. I know that as when I turned the key there was power but wouldn't even try to start. I tapped the starter and had some one key turning the key until it started. My question is, can the flickering light be from the starter or should I also replace my alternator too?