Per the headline, whether you've made payments in full every month is NOT reflected in your credit score or report. Instead, your credit report shows whether you've made at least the minimum payment on time each month  (and your credit score only reflects information in your credit report).
Hence, while a bad credit score is reflective of poor financial history, a good credit score does not reflect whether someone is paying off their credit cards in full each month or paying lots of interest by only making minimum payments .
What's In a Credit Score? 
This is 35% of your score. Interestingly, banks don't usually report late payments until they are at least 30 days late.
This is 30% of your score. The percentage of available credit that you're utilizing - not the absolute amount owed - is the metric used. So if you have $50,000 of credit available through your credit cards, and you've got a credit card balance of $2,000 across all of your cards, then your utilization is 4%.
Generally speaking, for credit cards the statement closing balance (NOT the difference between the statement balance and your payment) is used in the credit utilization calculation, though it can vary by bank. You can find a crowdsourced list at Doctor of Credit.
It's important to note that credit utilization is merely a snapshot in time, not some weighted average of past utilization history. Hence, if you are applying for credit soon, it is advantageous to lower your credit utilization. Furthermore, if you spend a lot of money with your credit card one month, yes your credit score will take a hit, but it will fully recover if you pay it off and don't spend as much next month.
Length of Credit History
This is 15% of your score. Generally speaking, the longer your history, the better. The most important metric is the average age of account, but there are two other metrics they use: the age of your oldest and newest accounts.
For the FICO credit score, which is commonly used for credit cards, closed accounts count towards your average age of account. However, they drop off your report in ten years. Hence closing an account will not affect your FICO score in any way for ten years.
On the other hand, VantageScore doesn't use closed accounts for calculating the average age of account.
So long as there's no annual fee to keep an account open, you should keep it open.
This is 10% of your score. You'll have a better score for having a variety of credit accounts (credit cards, mortgages, car loans, etc).
This is 10% of your score, and comes from inquiries. Every time you apply for credit, you get a hard inquiry on your report, which lowers your score. It stays on your report for two years, but stops affecting your score after one year.
What Happens to Your Credit Score When You Open a Credit Card?
- The hard inquiry on your report drops your credit score
- If this is your first credit card, it increases your credit mix and credit score
- So long as you keep your spending on credit cards constant, it will decrease your credit utilization and increase your score
- It will reset your age of newest account and decrease your average age of account. However, the more accounts you already had open, the less your average age of account is affected. This will lower your credit score.