I am a new landlord, and last week I filed my first Summons for Unlawful Detainer (I'm here in VA, which I've heard is landlord friendly).

My LLP (landlord policy) has very adequate coverage, fair market rent coverage, insurance on a 3 car detained garage.

The legal proceeding is called a Forcible Entry and Detainer action. 

I am unsure about Texas law and you should really consult a decent real estate attorney to assist you with this if you are unsure which parties to include on your forcible detainer petition.

An Illinois landlord would be under zero obligation to re-negotiate anything, and could treat any failure to comply with the terms as a breach, subject to relief from a forcible entry and detainer action (Eviction). 

@Rusty Whitney There is the Illinois Forcible Entry and Detainer Act, and the Security Deposit Return Act.

@Kenyo Henry  The legal way is file a Wrongful Detainer.

You are better off refusing partial payments, serving the 5 day notice, and then going forward with forcible detainer summons.  

I have seen the info online for example it says "unlawful detainer" it has plantiff and defendant, but it doesnt show you the address.

In Calif, if you accept even $0.10 towards Rent Due, then you will have your UL(unlawful detainer) case dismissed.