You've run into "progress" :)
For some decades US bathroom and kitchen faucets usually had brass metal male threaded inputs for their cold and hot water supply. These were usually 1/2", but sometimes 3/8".
These were then connected to the water supply shutoff valve (normally 3/8" male "compression", but sometimes 1/2") with flexible braid-reinforced supply hoses. Those flexible hoses have female connectors on both ends with rubber gaskets inside, making for easier leak-free connections and replacement if necessary. They are available in lengths from less than 12" to more than 6 feet, and with 3/8" or 1/2" or one of each connector on the ends.
In the never-ending quest for cost reduction and installation simplification, faucet manufacturers now make many of their faucets with braided flexible supply hoses permanently attached. This makes connecting each water supply a simple matter of securing one rubber-gasketed nut to the water shut off valve's standard compression-style male outlet (the ferrule and compression nut are not needed). This is what you have on your setup.
However, the water purifier kit seems to be made for the previous common situation, replacing a flexible supply hose in the middle, and comes with a real 3/8" compression connection for the shut off valve end, and a fitting for the post-filter end that would screw directly onto a standard male 1/2" faucet cold water input.
Your proposed Solution 1 - the push-on 3/8" to 3/8" male threaded compression end - seems the best one to me. The more fittings/connections/adapters you add, the greater the chance for a leak. As long as the end you've identified is made by a reputable manufacturer with a good reputation on push to fit fittings, you shouldn't have any issues. Be sure to read and carefully follow the installation instructions for both push to fit fittings.
PS - Also note that the rubber-gasketed metal connectors should be screwed on only to the point of "finger tight", then perhaps 1/4 turn further with a wrench. Tightening past this point can force the rubber gasket out of position and actually create a leak.
