Just listened to the video.
Low battery.
What you are hearing is the solenoid getting insufficient voltage to work properly.
When you turn the key, voltage (and current) is sent to the solenoid. The coil inside the solenoid is energized and a slider arm moves the spinning starter armature (which receives voltage at the same time) into the flywheel teeth on the ring gear. If all works properly, the starter turns the flywheel, which turns the motor and starts the car. When the engine starts, the flywheel turns faster then the starter can and the starter teeth are pulled away from the flywheel by a springy thing. This springy part is called the bendix. If the bendix was bad, you would either hear grinding as the starter teeth can't withdraw or a spinning sound as the bendix can't engage. From the sound of the video, your flywheel gear is safe. Which is something to give you relief. To replace the ring gear you have to remove the transmission.
Back to the problem.
The solenoid is not getting sufficient voltage to work properly. So, even though the battery is new, I would take it to be tested. Do a voltage test and a current draw. Also check for the individual density of the liquid in the cells if it is an open top battery. You might have a weak cell. There are ways to test for weak cells in sealed top batteries too just not as easy. Otherwise just do the first two tests. I would not just jump start the battery. It's easy but doesn't tell us much.
After you've proved these components to be good, then you can check for voltage leaks.
If the battery is good. Clean the battery terminals. Look for the cable that goes to the solenoid. I don't know where Volvo puts it so you may need to follow the cable or look at a manual. Once you find it, take the positive cable off the battery. Remove the cable going to the solenoid and clean the terminals. If you happen to have an ohm meter around, check the ohmage of the cable. Zero out your ohmmeter, unless you have a digital one. I can't tell you exactly how many ohms your cable should be, but understand how cable is rated for resistance. Cable is rated for resistance at 20 deg C (about 68 deg F) and 1000 Feet. Your cable is, what, 3 ft long? So for a 6-8 guage wire, the resistance per 1000FT is 0.4 to 0.6 ohms. So you obviously don't want to see much above zero.
Ok. So the cable's good. Now what? How dirty do you want to get?
The next step would be to remove the starter and do some tests on it. I pass on that for now.
Basically if all these parts pass, you can ignore them as the problem. A bad alternator DOES NOT decide if the battery is good or not. A bad alternator just doesn't replace the charge the engine removed from the battery.
Simple test for a bad alternator? Glad you asked. A good battery with no load should be about 12.5-13 volts across the terminals. I know you can't, but start the car. Open the hood and check the battery voltage while the car is running. What is it? It should be around 14.5 volts. It may be a little more if the car was just started, and it may be a little less if the car was running for a little bit. So voltage not much more than 12.5-13, probable bad alternator. Voltage over 16, also bad alternator. It's running 'away' and will fry your battery killing it.
