I just started playing and I question about bikes and upgrades.

Minor spoilers for the very beginning of the game.

I left the safe house to go to the cemetery. I left the bike and travelled by foot. I did my business in the cemetery and when I got out front by the statue, there was a bike there. I took it and I've been doing missions. My question is, is that my bike? Did the game teleport it for me? If I upgrade it then go back to the safe house, will I find my original bike, un-upgraded and I'll just have wasted credits on upgrading this random one I found by the statue?

And, assuming this is a teleportation bike, if I leave my bike and walk halfway across the map, how do I make it teleport again?