Battery or Alternator?

If the generator was not functioning, I don't think you would have been able to drive for 45 minutes. The battery would have run down before that. To answer your question about the new battery, no it will not be damaged, but it too will run down, typically in less than an hour of driving. That doesn't hurt the battery, but you will have to charge it again.

I should mention too that if the battery is really run down, it can take a good 15 to 20 minutes on a charger before it starts to recharge. It can take that long for the acid to become conductive and allow current to flow through the battery. The best is to charge it at a slow rate for a couple of hours.

If the old battery really is defective, the engine will still run on just the generator with a jump-start, but you won't be able to restart it if you turn it off or if it stalls. To test the charging system requires a professional load tester, but you can start the process yourself with an inexpensive digital voltmeter. With the engine off, a good, fully-charged battery will read 12.6 volts. More importantly, once the engine is running, the battery should measure between 13.75 and 14.75 volts. If that is what you find, the charging system is working, so it's just the new battery you need.

Here's an article that explains this better:

https://www.2carpros.com/articles/how-to-check-a-car-alternator

This one shows how to use a voltmeter:

https://www.2carpros.com/articles/how-to-use-a-voltmeter

You can find a perfectly fine one at Harbor Freight Tools for around $7.00. Walmart and any hardware stores have them too. I can help you set it up if necessary.

It's important to note that this is just a preliminary test. If it passes, it only means it is okay to perform the rest of the system tests, but that requires a trip to a repair shop with the proper equipment. In general those additional tests are not mandatory unless you're still experiencing some kind of symptoms after the new battery is installed.

These articles will also provide some useful information:

https://www.2carpros.com/articles/how-to-jump-start-car-battery-using-jumper-cables

https://www.2carpros.com/articles/symptoms-of-a-bad-car-battery

https://www.2carpros.com/articles/car-battery-load-test

https://www.2carpros.com/articles/how-to-replace-a-car-battery

Also check out this video:

https://youtu.be/Z-p4dMJUpUw

Of special importance, watch the polarity of the cables when attaching them to the new battery. Your car won't have a lot of unnecessary computers, but at a minimum, connecting the cables backward will cause a lot of blown fuses. The positive post on the battery is fatter than the negative post, and the cable clamps are different sizes to match those posts. There's plus and minus signs on the battery for the positive and negative posts. If you aren't sure of the cables, follow them and you'll find the negative one bolts to the engine, and a smaller one runs to the body sheet metal. Way too often we read here that someone connected the battery cables or the jumper cables backward, and now they have all kinds of expensive problems. That's why I'm mentioning this.