If you are leasing, then you have zero equity. Leasing is another word for renting. Renters do not build equity because they have no ownership interest in whatever item they are renting.
The lease on a vehicle is typically calculated as the current value minus the estimated value at the end of the lease period (plus any applicable finance charges and some variable fees). At the end of that time, the dealer is betting that the car can be sold for at least as much as the estimated value.
If the dealer took your trade-in at $14,000 and gave you a lease on a new vehicle for $3,000 additional, that means that the difference between the original value of the vehicle and the value at the end of the lease period should be about $17,000, which is the total amount you would pay. At the end of that time, the vehicle may be worth anywhere between $0 and $1,000,000,000 (hyperbole for effect) but no matter what it is, you don't get any of it.
Most lease agreements will specify a "buy-out" amount that you can pay to purchase the vehicle at the end of the term, which should be the estimated value from above. Your hope is that the buy-out cost is lower than the actual value. Your buy-out amount at some point prior to the end of the lease would be that amount plus any amount you still owe on the lease (since that amount is needed to cover the full price of the vehicle).
Based on your figures, it seems likely that the original value of the car was around $32,000, and you would pay around $14,000 (up-front) + $3,000 (over 3 years), at which time you either pay an additional $16,000 (the estimated value of the car after 3 years - depreciation of roughly half, which is reasonable) to buy it outright, or return it and owe nothing further. Right now, with a year left in your lease, you owe around $1,000 on the lease. If you wish to own the car, you must come up with that $1,000 plus the $16,000 (and assume that your lease agreement lets you buy out early). The fact that the car is worth $20000 right now doesn't have any bearing. In that case, yes - you would end up paying $16,000+$14,000+$3,000, or $33,000, to buy a $32,000 car - or very nearly exactly what you would have paid if you'd bought it outright instead of being lured by an $80 lease payment.
You say you want to "protect your equity" if the vehicle were totaled in an accident. You should already be carrying collision and comprehensive insurance on your vehicle; I guarantee the lease agreement insists on this to protect their equity. If the car were totaled, your first obligation would be to pay the financing company for both the remainder of the lease, plus the buy-out value of the vehicle (which they could no longer sell since it's been totaled). You are unlikely to have anything left over from insurance but you might be lucky.
Leases on vehicles which hold their value well over time end up being relatively inexpensive - a Toyota Corolla bought for $20,000 can be had used for $13,000 after 3 years, depreciating only 35% - it isn't worth that much less than a brand-new one because people trust them. Conversely, leases on cars that do not hold value well can be very expensive because you are still paying for the car. A 7-series BMW with an original price of $115,000 is worth only $45,000 after 3 years, depreciating 61%. You can pick up off-lease luxury cars fairly inexpensively because of this. (Don't assume that everyone you see driving a Bentley is rich.)