The exact liability an airline will have for lost or damaged baggage is dependent on the airline (specifically, its Contract of Carriage or equivalent document) and the applicable laws of the country/countries involved.
In general, airlines are indeed liable for both the bag and its contents. For the specific case of easyJet, their liability policy says:
16.5 Damage to Baggage
The following conditions apply to all carriage of Baggage by Us:
16.5.1 in respect of Hold Luggage, We shall be liable to You for its destruction, loss or damage during the time it was in Our charge and to the extent that damage did not result from the inherent defect, quality or vice of the Baggage;
However, their baggage policy says (emphasis theirs):
19.4.11 You should not include in Your Hold Luggage fragile or perishable items, money, jewellery, precious metals, silverware, computers, electronic devices, negotiable papers, securities or other valuables, business documents, passports and other identification documents or samples and we accept no liability for them save as stated in Article 16.5.3 (Baggage, Damage to Baggage).
Oddly, Article 16.5.3 specifically refers to cabin luggage, not hold luggage:
16.5.3 We are not liable for any damage to Your Cabin Baggage to the extent caused or contributed to by Your negligence;
I suspect they meant to say Article 16.5.6, which says (emphasis mine):
16.5.6 We are not liable in any event in respect of loss or damage to Baggage which is not permitted to be carried pursuant to Article 20 (Dangerous Goods) or for any fragile, valuable, perishable articles or articles not packed in suitable containers that have been packed in Your Baggage contrary to the requirements of Article 19.12 (Baggage, Items Unacceptable as Baggage).
I would assume that Article 16.5.6 constitutes the 'special rules' they referred to in their response to you.