As with the Christian sites, I think it would be worthwhile to subdivide the CLs into the four official categories. We already have a connection for each of them.

Clearly defined cultural landscapes (12):


Aranjuez
Bergpark Wilhelmshöhe
Garden Kingdom of Dessau-Wörlitz
Kew Gardens
Lednice-Valtice Cultural Landscape
Medici Villas and Gardens
Muskauer Park
Par force hunting landscape
Persian Garden
Singapore Botanic Gardens
Sintra
Studley Royal Park

My selection:

Medici - beautiful Italian gardens
Persian garden - best example of non-European garden design? (apart of Japanese and Chinese gardens which are not represented)
Sintra - iconic, or as Solivagant says: visually beautiful
Singapore botanic gardens - best representation of a botanic garden with non-European flora
Kew Gardens - represents the English garden

This another (sub) category where we can find good examples of very similar WHS in another category - or in this case 2 other categories -
"Secular Structure - Palaces" and "Secular Structure Park or Garden"
I can't work out why it is that these gardens have been categorised by UNESCO as "Clearly defined Cultural Landscapes" whereas many other, apparently similar, landscaped parks/botanical gardens etc etc have not! Some, I suspect, were inscribed before CLs were introduced or became "fashionable". Whilst some of this list climbed on the CL bandwagon to improve their chances and hide their weaknesses (I think of Aranjuez)!

So I am reluctant to even consider those sites on the list which are primarily European "Palace" or royal/ducal gardens since there seem likely to be better elsewhere - I include Sintra in that list. the Medici Gardens however are probably not going to be surpassed?

Singapore and Kew both represent "Botanical Gardens" (the later isn't really an "English Garden" surely?) and we will come across others of these also (e.g Padua, Rio). I am satisfied that both would still come through on the basis of their geography - but it might be worth double checking that we are going to be happy to duplicate or exclude the others?

There are Persian (or Char Bagh) gardens beyond this CL list as well. What about Shalimar for instance? Do the Persian ones have pre-eminence?
Possibly?

I completely agree that this category is arbitrary. I think the best way to go about these categories is to keep in mind if there are comparable sites in other categories that are better and just suggest the good ones.

My selection:

Persian Garden (same reasoning as Assif)
Sintra (apart from being visually beautiful, it also encompasses a site linked to esoterism which is quite unique)

Other gardens I feel offer nothing the little green man won't experience first-hand from the selection of natural whs or palaces/castles and their gardens

I would support Sintra too.
I've put my opinion on the garden sites in that thread.

My selection

Bergpark Wilhelmshöhe
Medici Villas and Gardens
Persian Garden
Sintra

My selection:

Garden Kingdom of Dessau-Wörlitz
Kew Gardens
Medici Villas and Gardens
Persian Garden

Here's the full Human Activity, Cultural Landscape category link for those who need it. This category has a lot of sites repeated from the landscapes/gardens category, but I'm going to select the ones in this category more along the lines of a cultural landscape:

1. Persian Garden -- diverse examples of Persian garden styles representing different climates and regions; had an impact on garden design from India to Spain
2. Medici Villas and Gardens -- displays unique style of Italian gardening, architecture, and landscape design, influencing other residences in Italy and Europe
3. Sintra -- well-designed ensemble of palaces and gardens in a mountain setting near Lisbon, Portugal; covers a variety of architectural styles, from Moorish to Gothic to Renaissance; influenced the spread of Romantic architecture in Europe

If ever I was going to pick Par force hunting landscape (as an example of an intentionally designed hunting landscape) it would be here, and yet, I just can't bring myself to do so. Maybe if I had a greater appreciation for the impact of Danish hunting in world culture...

Wrapping this one up:

Aranjuez
Bergpark Wilhelmshöhe
Garden Kingdom of Dessau-Wörlitz
Kew Gardens (already in on different category)
Lednice-Valtice Cultural Landscape
Medici Villas and Gardens - unique style of Italian gardening, architecture, and landscape design, influencing other residences in Italy and Europe
Muskauer Park
Par force hunting landscape
Persian Garden - already in on different category
Singapore Botanic Gardens - already in on different category
Sintra - visually beautiful, linked to esoterism and covers a variety of architectural styles, from Moorish to Gothic to Renaissance
Studley Royal Park