Exchange contract or signed on the same contract?

I have this dumb question to ask. Hope u don’t mind.

I had a bad experience before while the vendor throw a very low price , after I signed my copy , she didn’t sign  but instead use my offer to enforce other buyer to pay more and leave me on hook as the bottom gurantee  until the last moment when she signed with the higher offer.  I hated this so much that I swear I never sign it unless vendor signs also on the same piece of paper and I gotta keep a copy of that as proof.

Hence, I have done this ( i mean both parties signed on the same contract, and I got to have a copy of it. ) for the past two IP I purchased.

Now I purchased my 3^rd IP.  Instead of faxing me the both signed contract front page which REA had promised,  She then told me she had mailed the vendor signed copy to my solicitor.  Which later my solicitor had confirmed received and exchanged.

After I told the agent and Solicitor of my ‘normal’ way of dealing ( both sign on the same contract and obtain a copy of that), they think I have no idea and not follow the right procedure.  But I think my way of dealing is more sensible and more secure plus I have done the same twice before.  Now my question is:

Is my ‘normal ‘ way of dealing not correct and invalid from the legal point of view ? ( I mean despite both parties’ signature , but contract still regarded as not exchange until both solicitors received the original document via mail ? )

If as agent and solicitor said, then i’m back to the original status, ie. I have no way to stop people from using me but wait and hope they will do what they promise?

hope to hear your advise.