First off, the battery issues you describe could indeed indicate a battery problem. There’s a community-written battery guide here. Check whether your battery is swelling - see the first point under Problems and Solutions. If it is, it’s likely covered by warranty. You may want to contact support directly about your battery even if there’s no evidence of swelling.
Safe mode can be tricky, the easiest way for me is via the shutdown menu when the phone is on: press the power button until the shutdown menu appears, then press on the Power Off option until it asks to Reboot. Tapping OK should reboot into safe mode.
The other way you refer to is indeed not worded in the most straightforward manner (‘again’ refers to the switching on, not the holding of buttons). Basically, when the starting the phone, hold volume up and volume down at the same time until the phone has finished starting. I personally don’t like this method, as you end up in different system areas if you accidentally hold only one of the volume buttons.