When a contract is signed there can be no doubt about this but 'witnesses' can be 'bought', deny event, etc. That is why signed contracts are universally used throughout the world, so why would you consider the Qur'an saying any differently.
This does not apply to your case, but there is or was a growing trend in UK to just do 'nikkah' with mulla and no UK marriage registration. This has led to some men divorcing wives with no consequences as no official marriage occurred, they were just living together according to UK law; bad for women. Now some mullas have stopped nikkah only marriages and only do this if proof of British registration is there.
Back to your case, two witnesses, no signed contract reminds me of a Bollywood film where young muslim girls marries boy in front of his friends and 'bought' mulla, is used and thrown away next day.
You cannot do a 'do-it-yourself' nikkah, it must be officially registered.