Lifting the connector and simultaneously sliding a thin piece of card or plastic under the pins should be done carefully. In fact, to minimise damage I just slid it under ONE of the main battery pins, plus or minus, which are in the middle and quite robust compared with the tiddlers on the side of the connector.
The second point, the battery and the motherboard have protection circuitry. When I broke my connector the battery dropped to 0.5V at the terminals. Connecting the charger, when all the connections were sound and not shorted raised it from the dead back to over 3.2V
From a repair point of view. This connector is a royal pain. For assembly it's great, well, nearly great.
This connector is SOLDERED to the motherboard side. Never lever this on the motherboard side, I will call it the right hand side. It will break leaving it's soldered pins behind. Yes I did it.
The pins, on the battery side of this connector, just rest on the batteries plated pads. They are kept under tension by the screw. It should be noted though that removal of the screw will not disconnect the connector pins from the battery pads. Natural tension is still present.
I suggest strongly that you take a look at the battery connector online to see the nature of the pins and you will see that some are quite delicate.
The tools purpose is to raise the battery side of the connector ; the left side, so there is some clearance between the pins and the batteries pads. If you have seen a picture of this connector then you will realise the potential of damage to this connector and will have a better idea of what to do.