The language in your note sounds soft and ambiguous.

I would like to know your experience with this and how to help secure this tenant when you have a language barrier. 

I say 99.9% because I don't represent 100% of the insurance carriers in the US, so there may be one out there that has unique language in thier policy, but this is the standard practice.

This is the language on the contract regarding duration Duration of Agreement.

I've changed my language and docs to not include the word 'pet' anymore at all. 

Especially one who speaks a foreign language?

If you do decide to allow pets, be sure to have language in the lease that protects you financially if the pets cause bodily injury or property damage to you or anyone else. 

All payments made by Tenant to Landlord after the tenancy commences, no matter how designated by Tenant, will be applied as follows: First to any outstanding amounts for damages/repairs, utilities, etc.; second, to any outstanding service charges and fees from prior months; third to any rent outstanding from prior months; fourth, to any service charges or fees due in the current month; and lastly to the current month's rent."

If your goal is for tenant's to reach out to you before calling a repair man, you can probably draft language requiring the tenant contact you for repairs prior to contacting a third-party repairman.

Look for body language too when you ask this question.