The rules for obtaining a residence permit on the basis of an employment contract have not fundamentally changed (you can read about the conditions
here). The version of the residence permit for labor migrants has been updated. Most likely, the package of documents for obtaining a residence permit will now include a notice of registration of a foreigner as a labor migrant with the Ministry of Labour, Health and Social Protection. The form itself is not specified in the amendments; possibly in the form of a certificate or an extract from the registry.
It should be reminded that the draft law adopted in the third reading will be sent to the President for signature within 10 days. The head of state has 2 weeks to approve it or return it for revision. The Law shall enter into force on the dates specified therein.