The statement in grey is not right. Sanctification DOES require work on our part.
I believe it follows on from what you highlighted as being correct.
The salvation rest being from resting from our works in order to earn Gods favour.
Yes sanctification does require effort on our part.
No doubt about it, but not too earn or keep Gods favour.
So we have been sanctified, which means set apart.
Made Holy and to grow in holiness. To do good works per Eph 2.
Jesus said
John 17:19
19 And for their sakes I sanctify Myself, that they also may be sanctified by the truth.
Peter (the hot headed man, who denied Jesus, cut a guards ear off, who could only respond when Jesus asked him 3 times "Do you Agape me could only respond with I Phileo you) said
1 Peter 1:15-16
15 but as He who called you is holy, you also be holy in all your conduct, 16 because it is written, “Be holy, for I am holy.”
Hebrews 2:11
11 For both He who sanctifies and those who are being sanctified are all of one, for which reason He is not ashamed to call them brethren,
1 Peter 1:2
2 God the Father knew you and chose you long ago, and his Spirit has made you holy. As a result, you have obeyed him and have been cleansed by the blood of Jesus Christ.
May God give you more and more grace and peace.
So who is it that sanctified us?
Do we sanctify ourselves?
Who is it that makes us holy
Ourselves or the Holy Spirit?
So to me to say that our works keep us saved puts the ball back in our court.
If thats the case it takes the focus off what Jesus has done for us then we feel we need to add to it.
We have nothing to add to what he has done.
When we accept what he has done and allow the Holy Spirit to work in us, he will.
Not to be saved but because we are.
If a person does not believe they are saved there is only one path to walk.
That is the path of works.
Just like the Muslims do. Hope their good works outweigh their bad works. (not saying you are a Muslim)