It's true, when I've not exercised for a while my body starts to hurt and I can feel my body going back in terms of my muscles shrinking.
The feeling of a muscle strain is very different to DOMS but the tightness and stiffness you feel with DOMS is similar to the tightness you feel when an injury is developing, like pre-injury.
If I feel some minor strains I normally would stop. I'm a little careful if I train too much or too hard.
The minute something doesn't feel right I stop. I would rather lose one day of training making sure I was okay than powering through it and being out for weeks or longer. Most of the time I know I'm fine, and i could totally work through it but I like being on the better safe than sorry side of things.
That's probably the most sensible option to take but I have never had a problem training through minor strains, especially when it is a Friday and I know I can rest through the weekend. I rarely get any strains or injuries, though.
It all depends on the type of strain and the location. If I am having a wrist strain, I have no issues working another body part obviously. I have been dealing with hamstring issues over the past 5 days and just aggravated it today doing lunges. I actually thought that it was fine. I did squats with full weight and even leg curls with full weight, but on my second time down on the lunges, I fell to the ground. Luckily, nothing popped. I iced it down and will rest it up for a while again. Stupidly, I have been dealing with tendonitis and fighting through it. I know that I should be stopping, but I already deloaded for a week and it did not alleviate the pain. I have learned to alter specific exercises so that my elbows are not hurting during the lift. I know that my issues have more to do with age as well and I am getting older and still lifting to add mass like I did 20 years ago.