I would hesitate to deem any reasonable technique "correct" or "incorrect." The methods of serving you have described are both equally valid; what works best for a particular player depends on their skill, experience, athleticism, and ultimately personal preference.
You've noted, and I agree, that most professional players' motions can be best described by option two (thrusting upward and off the ground into making contact with the ball). As noted in the previous answer, they use this motion because it is the most effective. That said, this kind of service action is more complicated to achieve. It requires coordinating your jump with tossing and contacting the ball, which takes practice. I would call this a "traditional serve" rather than a "jump serve" for reasons below1.
The first option is simpler, and easier to learn, since there are fewer moving parts. You don't have to worry about timing an extra element (i.e. the jump). For this reason I have seen plenty of less experienced players using something similar to option one. I would recommend someone who was just learning to play to start with a simple motion (i.e. no jump) and then work their way towards a full-fledged professional-style service action once they feel comfortable timing the simpler serve.
1 Note that what comes to my mind when I hear "jump serve" is something more extreme like this -- yes this is actually the way one professional tennis player serves.