Interest: what is the most strategic time to make a purchase: just before or just after the statement comes out?

In order to pay a lower interest rate on my credit card balance...

pay the balance in full by the due date. That way, you pay $0 interest. No need for scheming and strategizing: if the bill says $893.61, then pay $893.61.

(To sanely manage your budget, I recommend you pay your current month's charges at the end of that month, no matter when the bill is actually due. The Internet makes that trivial.)

It won't make a significant difference. You're most likely* charged interest on the average daily balance over the statement period, so if $200 of that balance is only on your statement for a few days it won't contribute very much to the amount of interest owed. If you pay off the complete balance before it's due, then you pay ZERO interest and don't have to worry about such scenarios.

I would be remiss in also suggesting that if you are adding to a balance that you haven't paid off yet, then you are just compounding your problem. I would strongly suggest that you stop putting any more charges on the card until you can get your spending under control so that you can buy items with CASH and stop wasting money on interest. That $200 purchase will become a $250 purchase if you continue to carry a balance for another year.

*Check the terms of your card to be certain, but every credit card I've seen works this way.

tl;dr: Prolong every new charge as long as possible until you are back to paying in full every month.

In the best case scenario, if you are paying your balance due in full every month, it doesn't matter when you make a purchase as you don't pay any interest. In your case though, since you are currently carrying a balance, then the answer is always the later you can make the purchase, the better. Note this means the statement date is not relevant in this case. The reason is because the average daily balance will always be lower on any statement the longer you wait to make a purchase. In fact, you can extrapolate this to conclude that you should prolong any new purchase on the card literally forever until you are paying in full again every month. Therefore, if you have the funds I would recommend using cash or a debit card until your credit card is paid off in full. At that time you can start using the CC again.

Tip: most credits cards lose the grace period once you carry a balance, and they require you to pay in full for two complete billing cycles before the grace period comes back and interest is waved again. This means that once you have enough to pay in full again, you should pay the entire balance off, not just the "statement balance" on that bill. Once you do that for two months in a row, then you can start using the card freely and paying just the "statement balance" again every month without paying any interest.

As @DStanley notes, most credit cards -- every credit card I've ever had -- charges interest based on average daily balance. It doesn't matter when the closing date of the statement is. What matters is how many days the charge was on the card before you paid it. So waiting before buying something on the card will save you money, because that means more days that it adds $0 to the balance (because you haven't bought it yet).

Better still, pay off the card in full every month and pay zero interest.

There is one way that the closing date of a statement matters. If you pay off the card in full, you pay zero interest. But if you don't pay it off in full, you pay interest on the average daily balance.

So for example, to take a simple case say that you bought one item for $100 on the first day of the billing period and that's all you bought all month. You get a bill for $100.

Scenario 1: You pay the full $100 before the due date. You pay zero interest.

Scenario 2: You pay $20, leaving a balance of $80. You pay interest on the full $100.

Scenario 3: You pay $99.99, leaving a balance of $.01. You pay interest on the full $100.

I'll add the caveat that the terms of your credit card may be different, but I believe that's how it usually, if not always, works.

If you're NOT paying off your credit card bill in full every month, I'd strongly encourage you to start doing so. Credit cards are a very expensive way to borrow money. Instead of buying on a credit card and then spending months paying it off, save up for months and then buy with cash. (Or buy with a credit card and then pay it off immediately.) Sure, if you have unexpected expenses and find that you need to put the rent on the credit card or you'll lose your apartment, or you need to put food on the credit card or you'll starve, okay. But most things that most Americans buy on credit are far from necessities of life. Put off buying that new bigger TV or game console or designer shoes or whatever until you can afford it, and in the long run you'll have more money.

Credit cards that carry a balance almost always charge for new fees on an effectively daily basis.

Credit cards that do not carry a balance (are fully paid off by the statement due date) almost always do not charge interest.

If you are able to fully pay off your credit card for the period leading up to the new statement being issued, but unable to pay it off if you added on the new charge, then charging the cost after the statement comes out gives you an interest grace period.

If, however, you are already carrying a balance, then the "statement date" doesn't have a discontinuous impact on the total interest you'll be charged. The effect of delaying an additional debt will cause a reduction in debt carrying cost (interest), but the specific statement date will just mostly just defer when you are told about the interest on the new debt, and have a very small impact on the amount of total interest you'll owe on that debt.

Concrete examples.

You know you'll own 100$ on a credit card this month. Last month you paid it off in full, and have a new 1000$ cost. Your credit card charges 20% annual interest (1.5% per month, 0.04% per day).

You cannot afford to pay off the 1100$ when your next statement is due, but you can on the statement after that.

If you charged before your next statement, then carried it for a month, then paid it off, you'd pay a total of 1115$.

If you charged it after your next statement, paid off the 100$ on the next statement, then 1000$ on the one after that, you'd pay a total of 1100$.

There is a discontinuity here of 15$ from moving the date of debt by a day or two, far greater than the interest on 1000$ over 2 days (which is about 80 cents).

Now, suppose you have a 100$ credit card debt you didn't pay off last month. You'll owe 101.5$ next statement if you don't charge. If you charge the 1000$ the day before the statement, you'll owe 101.5$ + 1000$ + 0.4 on your next statement. Then another month passes, and you owe a total of $1118.43.

If you instead charged the 1000$ the day after, you'd owe 101.5$ on your next statement. Then you'd owe 1117.62$ on the statement after that.

The difference -- 80 cents -- is basically 2 days of interest on 1000$. There is no "discontinuity" due to the statement date here.

Note that the terms and conditions of your specific credit card can be different than what I modeled above.