What does aman mean when it doesn't mean “faith/belief”?
But Moses' hands were heavy; and they took a stone and put it under
him, and he sat thereon. And Aaron and Hur held up his hands, the one
on the one side and the other on the other side; and his hands were
steady (emunah, אמונה) until the going down of the sun.
-Exodus 17:12, KJV
The root used here is aman. In general, the word aman means to make something steady by literally securing it in a physical sense.
What does batach mean when it doesn't mean “faith/belief”?
When batach is used as an adjective it means physical security and protection from attack:
Then the five men departed , and came to Laish, and saw the people
that [were] therein, how they dwelt careless, after the manner of the
Zidonians, quiet and secure (la'vetach, לבטח); and [there was] no magistrate in the
land, that might put [them] to shame in [any] thing; and they [were]
far from the Zidonians, and had no business with [any] man.
-Judges 18:7, KJV
When batach is used as a verb, a person is actively boteach in something else. In this case the root word batach means to have faith that the something else will provide physical security from attack:
And he shall besiege thee in all thy gates until thy high and
fortified walls come down wherein thou trusted, (boteach בטח) throughout all thy
land; and he shall besiege thee in all thy gates throughout all thy
land which the LORD thy God hath given thee.
-Deut. 28:52, KJV
And Gaal the son of Ebed came with his brethren, and went over to
Shechem: and the men of Shechem put their confidence in him.
The dual nature of “faith/belief”
Aman and batach are both used in the Bible to mean “faith/belief,” but the “faith/belief” which derives from the word aman is different from the “faith/belief” which derives from the word batach.
There are two participants in the act of belief – the believer and the precept or idea or being or concept which is believed in. The act of belief can be characterized by how it affects the precept which is being believed in and by how it affects the agent who is doing the believing.
If I choose to be ma'amin (from the root word aman) in the Pythagorean theorem or in God, I am privately casting a vote of confidence in that precept and defining myself among a community of people who support and uphold that notion. In saying I am boteach (from the root batach) in God, I convey that I am afforded security in that belief. (It wouldn't make sense for me to say I am boteach in the Pythagorean theorem.)
Clarification about the meaning of batach (in response to JonEricson's question):
When batach is not used in relation to God it means physical protection from a physical attack. When batach is used in relation to God it has a broader meaning which includes: "to rely on God" and "to find security in faith."