I definitely have spent some time becoming familiar with all of these things over the years!
Here is an article (on one of the web sites I mentioned) that explains the difference: How to Get Merit Aid from Colleges | CollegeData.
Think of merit aid as a way for schools that are not in the tippy-top to compete for amazing students like you. You can gauge your strength as an applicant and see which schools still fits all your interests and gives you a great deal (this just means you shop for schools wisely and do research to discover a school’s strenghts when you are not as familiar with it).
One thing that gets confusing is the idea that the best (and wealthiest) schools meet all of a student’s need. NEED is a relative term. When you say that your parents use their income to support family overseas, that means that there is actually less money for your education than a school thinks you have available. You can try to explain this to a school, during a financial aid appeal. However, it is not guaranteed that the expense would be considered a ‘necessary one’. (I don’t know much about it but if your parents don’t pay at least half of the relatives’ expenses, then the relative is probably not officially considered a dependent.)
So the FAFSA or the CSS Profile (learn which ones you have to submit at each school) may come up with a different expected family contribution (EFC). Side note: many schools meet 90-95% of the demonstrated need rather than the whole thing, so that can also be an issue. The EFC that you receive on the form is sort of a starting number rather than a final number.
Schools that meet NEED often don’t need to attract students because they are already so popular, so the bargains are at the school that may be out of the way, or not as well known.
You can begin evaluating a price fit by playing around with those net price calculators. Try one of your favorite schools, but also try schools you had not thought of before! Once you do one, many of the Net Price Calculators look very similar so it shouldn’t take you long after you gather all the information the first time. (Here is Boston University’s: https://www.bu.edu/finaid/aid-basics/cost-of-attendance/net-price-calculator/ and Clark University’s: https://www.clarku.edu/offices/financial-aid/prospective-students/u-s-students/net-price-calculator/)
Keep charts of all the information (okay, I am a sucker for a chart…) and you will be an expert college shopper in no time.