While I would always strive to be cordial and civil about these kinds of things, I don't think the difference in presentation would make any difference if the lawyers are involved. Either modifying the terms is acceptable or it's not.
While negotiating employment, one would think it would be natural for prospective employees to engage the services of a lawyer to ensure everything is aboveboard. After all, the company sure as hell had legal draw up and/or sign off on their contracts. With freelance contracts, of course both sides should have lawyers signing off to foster a fair relationship.
Yet what does an employer think if a candidate indicates that they have involved their own lawyer during contract negotiations? Perhaps it's a normal occurrence in major centres (ie: Silicon Valley), or maybe even across the US. I have never heard of a single one of my 1000+ peers over 15 years of experience having done so in my country (Canada).
The very act of engaging a lawyer to read through one's employment contract here would be seen as an aggressive attack against the "integrity" of the employer's business. Red flags galore raised about how this person would be a risk to employ, should they go running to a lawyer over every little grievance.
If I had a lawyer make amendments to my contract and sign off on it, and slipped to my prospective employer that this act had taken place, I'm quite sure my résumé would hit the bottom of the recycling bin within seconds of my name being added to a blacklist of "never communicate with this person ever again; do not reply to their emails, hang up the phone the moment they give their name, call security if they show up at the front desk, etc.". I would be unemployable if I even mentioned the word "lawyer". I suspect cultural differences, with the US being desensitized to the presence of lawyers in an abundance of everyday situations?
So, no lawyer for me. I much prefer the idea of requesting that the company amend the contract themselves to accommodate me, rather than red-inking it myself. Maybe Canadians really are too polite? ;)
tldr; In Canada at least, red-inking your contract on your own has a chance to succeed. But involving a lawyer would make employers avoid you as a high-risk hire.