Vehicle does not start immediately after turning key + seatbelt issues

Hi, I have a 2009 Suzuki Swift+ (which is Canadian only, but pretty much a carbon copy of the Chevrolet Aveo, which is why I am posting on this forum to see if any Aveo drivers have had similar problems). I've got around 18,000km on it, and have been having the following problem for about a month now:

When I put the key in the ignition and turn it, absolutely nothing happens. In exactly the way that it would if there were a dead battery. However, if I hold the key forward, the car will start without any unusual noises / other problems. Alternatively, if I put the key in, let it sit in the ignition for about 5 seconds, and THEN turn the key the rest of the way, it will start without a problem.
This problem occurs after the car has been sitting for 5 or more minutes. Immediately I thought it would be a loose battery terminal connection, but upon checking, I found the connections to be solid, and there is absolutely no corrosion on the terminals or the wires.
I first took it into the dealership last week, and they reset the computers, but the problem is still happening. There are no issues with the car's performance while it's running. I've taken it back into the shop, and am waiting to hear what they say, but am looking to see if anyone has experienced similar problems or found solutions.

Separately, I have had an issue with my seatbelt where, while shoulder checking, the seatbelt locks up and does not let go, only gets tighter. At one point, I actually had to take it off completely, and could not get it back on as I was driving down the highway. Only after about 5 minutes of trying to gently coax it did it relax enough for me to be able to buckle up again. I haven't had this problem in two months or so, and even then it was intermittent. Is this a problem I should get looked at? Or just a one-time 'let-it-go' sort of thing.

Any insights in either case are appreciated
-Jason

UPDATE:

First visit to Suzuki dealer: They reset the computer system, and sent me home with it. Problem immediately occurred again (starting up in the dealer's parking lot)

Second visit to Suzuki dealer: Their service department contacted Suzuki and were told about a Service Bulletin that was put out after a number of similar reports. The dealership followed the software reprogram steps and gave the car back to me. The first start looked promising, starting with no delay. As the day progressed and I ran more errands, the problem quickly became regular again.

I took it back to the dealership, where I was told there was nothing else they could do, since they followed the software reprogram codes. I told them that wasn't good enough, and asked if Suzuki had something in place to replace my vehicle with something that works. They tried to tell me that the problem was just a result of the way the software for the car is designed, and that's just the way the car is. I told them that's impossible because I've had it for 8 months where there is absolutely no delay on the starting. They said they'd speak to Suzuki on the phone next week to see if a further resolution can be reached.

This is starting to get frustrating.

its such a weird problem, idk what it could be myself, could you possibly make a video of the problem?

My 08 Aveo(bought last week and right to the shop for passenger airbag light remaining illuminated for an extended period and now waiting for the dealer to find a passenger side pretensioner) doesn't have this issue.

I did have a cavalier with a similar issue. That would happen basically because my fuel pump wasn't working well. I thin(it was a few years ago) that it was temporarily improved by fuel injector cleaner. Shortly after the fuel pump went completely out and the car wouldn't start.

Never heard of the belt problem. Good luck!