First of all welcome to The Automotive India buddy.
Now coming to the main point which car?
I would suggest you EON any day over the Alto 800 (personal choice) as it fulfills all your requirements perfectly and will be in your budget.
I have recently bought EON D-lite Plus, here is my ownership review: Hyundai EON (D-lite+) : Ownership Review
And regarding the shaky gear lever, i don't feel that in my EON maybe because it is a new car right now or maybe Hyundai has taken care of it. I was under the same dilemma when i was in the market looking for one.
I read all kind of reviews about EON and especially the shaky gear lever. So finally i went in and took a test drive of EON and Alto 800 and right away booked the EON. And yes i will recommend this car to anyone looking for a car in that segment.
It's a VFM
car from Hyundai from every angle.
But i'll still suggest that you go and take TD of both the cars.