EIDE will work AFAIK. EIDE, PATA, ATA, UATA, UltraAta, ATA100, ATA66 are all denominations of IDE flavours. The year of Your laptop (2004 or newer AFAIK) is pretty recent for IDE, I don't see a reason for it not to support any mode.
If You want to, You can post the model of the dead drive and I will look it up for You.
The drive You listed is ok. If You are a corporate IT technician and should repair a corporate Laptop (You wrote "ordered", which only bosses usually do) then I would go with that drive as it should be bigger than the original drive and is the cheapest in the list. Accountants look at every penny and I would assume that this laptop currently has a role that is matched to its former capabilities. (In Corporate I usually go with KISS philosophy {keep it simple, stupid} whereever I can if it doesn't hurt processes, because changes usually disrupt users/staff)
(As long as none of the others have any issues with the drive. I don't know of any issues with the model.)
Which OS will go on the laptop? If it is not for corporate but consumer they might be happy to pay a premium for a bigger drive.