The cold is an entirely different virus, unrelated to the flu actually. You can't get a cold from the flu shot, it's physically impossible. You can however always get a cold from infected people around you. You only get immunized for the flu, not colds.
Consider this: a cold won't kill you. Unless you're already very ill and not managing your diabetes well, there's virtually no risk in having a cold. The symptoms are usually sniffles and congestion, sneezing, sore throats, nausea, mild fevers if any at all and coughs, and start out slowly and build up over time. It can take a week before your uncomfortable throat has turned into a full on cold. You recover rather quickly from these.
The flu is a much more severely damaging virus that CAN kill you. Even healthy people have a risk of developing complications from it, so we're even more likely to get hit by them. Flu symptoms are similar to colds, but they include aches in muscles and joints, weakness and extreme fatigue. More severely you can have trouble breathing, pressure in the chest and sudden dizziness (which requires immediate medical attention). A flu hits you really hard very suddenly unlike a cold, and it usually takes 2 or more weeks to fully recover from because of prolonged fatigue.
I know shots are icky and needle phobia can make this a massive hurdle. I understand that people may not be able to overcome that. But if you can think for even only a second that you can take that quick prick into your arm, and your doctor has cleared you for it, go for it. We live in an age where a flu no longer means a death sentence precisely because we have vaccines.
Only a hundred years ago 50 to 100 million people in the world died from a massive flu pandemic. We don't have to worry about that anymore because of these shots, so I will never stop advocating for people to get them as soon as they can.