Maḥăsê was one of the most common, found for example in Psalm 46: 1 - The Lord is our refuge and strength, a very present help in times of trouble. It's also occasionally translated as shelter.
But twice in that same psalm, refuge is the Hebrew word miśgab, translated in other parts of the Bible as defence, or "high tower." It's used also in Psalm 9: 9 - the Lord will be a refuge for the oppressed, a refuge in times of trouble.
Mānôs is used only a few times, sometimes translated refuge and sometimes translated escape. There are scriptures that even use a few of these refuge words in the same verse:
Psalm 59: 16 - You have been my defence (miśgab) and refuge (mānôs).
Psalm 94: 22 - The Lord is my defence (miśgab) ... and the rock of my refuge (maḥăsê).
Then there's sēṯer, translated in Psalm 32: 7 and Psalm 119: 114 as, You are my hiding place. It's more often translated in the Old Testament as "secret," so it's referring to the kind of refuge where you can't be found.
I meditated on many verses and I discovered even a few more Hebrew words, but the conclusion is clear: Yahweh Machsi is our safe place, a hiding place; the place we can escape to when we need defending or protecting; our refuge and strength in any kind of trouble we might face.