Looks like you bought your LED
strip on ebay, huh? I get mine there too... they take the cut-offs left over from cutting prime reels off the bulk reel, then solder them together to sell as sub-prime reels. Nothing wrong with them, just that they are soldered together segments. I'd suggest if at all possible to cut it where it isn't soldered together for your connections, as the teeth may have trouble with the extra material present.
Looking at your second picture, your connector is for a wider LED
strip. I'm guessing if you measure your strip and search online you'll find the correct (narrower) connectors.