There's a whole lot of lighthouses in and around Sydney Harbour, but are any of them open to the public, meaning that you can go inside and (ideally) watch the light in operation? Guided tours would be fine, and I'm willing to travel a few hours up or down the coast if needed.
Since you've scoped your question to within "a few hours up or down the coast", there are several options. I'll limit myself to a 3-4 hours drive from Sydney CBD.
You already found the Macquarie Lighthouse, so I'll skip that one.
Situated just over an hour north of Sydney, in Ku-ring-gai Chase National Park (note: park entry fees apply), 91m above sea level. Built in 1881 from sandstone quarried on site. Its oil room and keepers’ cottages remain unpainted in the original stone finish.
Access is over sealed roads, suitable for 2WD vehicles, followed by a 1km walking track.
Guided tours (30 mins) are conducted every Sunday. Contact Kalkari Discovery Centre (9472 9300 or 9472 9301) for more information. Prices mentioned here (not sure how up-to-date those are): $3 for adults, $2 for children.
Photo by bilgolabob, courtesy Youtube.
Situated about two hours north of Sydney. Modelled after the Macquarie Lighthouse and completed in 1903.
The lighthouse is available for accommodation, tours and weddings. At the time of writing this, tours cost $6 per adult, $5 per child (6 to 15 years) and children 5 and under are free. For a group of 10 or more then a special price of $5 each applies, A family pass is also available for $20. Prices for accommodation vary with the season and weekdays vs. weekends (roughly $200~500/night).
Photo by Yun Huang Yong, courtesy Wikimedia Commons.
Situated about three hours north of Sydney. Established in 1872, now an inactive lighthouse without a tower, which makes it look more like a regular house. The light was shown through the window of a 10 ft frame lantern room attached to the keeper's house. The light was moved outside the lantern in 1995 and then discontinued in 2003.
The house is accessible by road from Nelson Bay. It is open daily for visits (10am-4pm). Includes a tea room, gift shop and heritage museum.
Photo by wood26, courtesy Wikimedia Commons.
If you don't mind venturing out further, there are a few more lighthouses that are open to the public; i.e. Montague Island Lighthouse and Green Cape Lighthouse (both along the south coast of NSW). The site you linked to in your question has a convenient map with lighthouses as POI:
Map courtesy of lighthouses.org.au.
I wasn't intending to answer my own question, but as ever I hit on the right Google keywords just after posting and found one place:
Photo by Adam.J.W.C., courtesy Wikimedia Commons.
Macquarie Lighthouse, in Vaucluse at the eastern edge of Sydney, is Australia's oldest lighthouse and still operating to this day. Once every two months, on Sundays between 10 AM and 4 PM, they have tours open to the public. Advance bookings are mandatory and the cost is $5/3 adult/child.