Sounds familiar, except for the part where your employer changed the words.
This offer also came with verbal assurances that it was just a formality so I should just sign it.
I offered them two potential solutions. I had my lawyer modify their docs so they'd be acceptable to me, and I also offered my own docs which are specifically drawn up to be just about as even-handed as legal documents can get (I sign regularly as both buyer and seller), but they didn't find either of the options satisfactory.
I'm somewhat disappointed that it didn't work out, because the work was very well suited to my background and the other team members seemed great; but if that's how management treated me while trying to convince me to join their team, god only knows how they would've treated me down the road.
After that fell apart, I wound up with a new primary client who was much more reasonable. They passed me lousy docs, and I asked if we could use my docs instead. After a reasonably quick review from their legal counsel and some minor tweaks, we were off to the races.