Ann: Dorado-1 Well Update, page-91

The Britoil 121 is just a small towage tug and would have 'wet' towed as you say. It would most likely have all foreign crew onboard and would therefore have to hand the tow over to an Australian manned vessel before the legs could be placed down. It's a bit of a grey area one I don't full understand and doesn't always seem to conform but essentially once the rig touches the seabed via anchors or legs the tow vessel would also be deemed to be in contact with the seabed and need to have Aussie crew onboard. Also Siem and Maersk run much more sophisticated/powerful vessel's which make positioning much easier.
I have not checked their positions recently but sites such as marinetraffic give very delayed information if any at all unless you are a subscriber.
They may have used the lee of the island to provide calmer weather for a handover of the tow wire. I doubt the current conditions would stop or slow the operation but that is at the discretion of the master.

I imagine the Maersk Master will assist in the positioning of the jackup as two tow boats are required to obtain an accurate position. One as the primary tow vessel whilst the other acts as the 'brake' vessel. They both exert x amount of tons in opposite directions on the rig to prevent the current and weather drifting it off location whilst it gets it's legs down.

As I hinted at earlier there will be other vessel's supplying the 107. Currently it will be a PSV until a later date. Being a PSV it cannot assist in positioning so would account for a Maersk vessel filling in.

Hope that helps