For us in the U.S. 3rd shots are for immunocompromised, boosters are given for those in the specific populations mentioned. They are being considered two different things because a booster may be your 3rd, 4th,5th, etc shot. An immunocompromised person could have a 3rd shot which is designed to bring the efficacy level up (as it is believed their immune systems may have prevented them from getting the protection the dosage was originally designed to get) but they also may end up getting a booster which would be their 4th, 5th, etc. Boosters would help bring the efficacy up for those with underlying medical conditions should that have impacted it and underlying medical conditions is part of the population that was given the go ahead to get a booster.
The timing of 3rd doses and boosters also is recommended at different times in the U.S. My in-laws ( a cancer survivor and the other with an autoimmune disease) both had their 3rd shot of Moderna. Theirs was technically about 4 and 4 1/2 months since they were considered fully vaccinated and those who need the 3rd shot are recommended to get it immediately. Those who are qualified under a booster (which at the moment only Pfizer is authorized for boosters) should wait til 6 months since their first dose.
But for this thread I actually wasn't talking about the purposes of travel, that has zero bearing on my comment. I said that for the purposes of the mixing conversation that was going on. In the U.S. for U.S. citizens they are not saying it's acceptable to get whatever mRNA vaccine. They want you to stick with whatever you've had for your initial doses. For those who were talking about that that part, IMO, seemed important, that our CDC has not given the go ahead for mixing even when it's for the purposes of its own citizens for the 3rd dose or booster. I know however they are starting to get data and look over it for the mixing aspect which may help eventually give peace of mind and more freedom of travel for those wanting to travel and are running into issues of mixing being recognized.