You do not have adjustable coilovers. You have OEM style struts with what look like aftermarket lowering springs. The color might indicate Vogtland brand which are a 1.4" lower setup. Like these:
shop.redline360.com
That suspension link in your 2nd picture is just an end link for the sway bar. Sway bar end links do not affect how low your car is.
If you want coilovers look at a quality brand like BC:
If you spend less than $1000 on adjustable coilovers... you shouldn't be buying adjustable coilovers at all. Maxpeedingrods and Racelands are dangerous and can break due to poor metallurgy.

Vogtland Lowering Springs Hyundai Tiburon (2003-2008) 950516
Fitment Information Hyundai Tiburon (2003, 2004, 2005, 2006, 2007, 2008) Specs & Features Lowers 1.4" Front and 1.4" Rear Includes all 4 lowering springs Works with factory shocks or aftermarket shocks that fit stock springs Vogtland Performance Lowering Springs are manufactured in Germany...

That suspension link in your 2nd picture is just an end link for the sway bar. Sway bar end links do not affect how low your car is.
If you want coilovers look at a quality brand like BC:

03-08 Hyundai Tiburon
BR Series "These images may not represent your application" The BC Racing BR Series offers the most affordable entry into coilover suspension without sacrificing performance, comfort, build quality, vehicle height, or handling. This is done by extensively testing each vehicle then refining the...
shop.bcracing-na.com
If you spend less than $1000 on adjustable coilovers... you shouldn't be buying adjustable coilovers at all. Maxpeedingrods and Racelands are dangerous and can break due to poor metallurgy.