When it's used with a body parts and the message is given toward no one specific, I feel that it has a nuance of "your OWN (body parts)". So, その目で確かめよう is like See it with your OWN eyes or See it with your eyes. both translations make sense. This is often used in a TV commercial, poem etc and the message is given to anyone, no one specific.

Some other times we might use そのX to mean "your OWN (body parts)" would be:

• （あなた自身の／あなたの）その手で試してみてください
• （あなた自身の／あなたの）その手で夢をつかみとれ
• （あなた自身の／あなたの）その耳で確かめよう
• （あなた自身の／あなたの）その目で確かめよう
• （あなた自身の／あなたの）その心で感じよう

The その in その手に触れるまで is very similar but it's used more as a simple "your" and it sounds like the speaker is thinking about a hand of someone specific like someone the speaker is in love with, etc. Unlike the examples above, the translation for this one would be until I touch your hand. It can't really be translated as until I touch your own hand.

When it's used with a body parts and the message is given towards no one specific, I feel that it has a nuance of "your OWN (body parts)". So, その目で確かめよう is like See it with your OWN eyes or See it with your eyes. both translations make sense. This is often used in a TV commercial, poem etc and the message is given to anyone, no one specific.

Some other times we might use そのX to mean "your OWN (body parts)" would be:

• （あなた自身の／あなたの）その手で試してみてください
• （あなた自身の／あなたの）その手で夢をつかみとれ
• （あなた自身の／あなたの）その耳で確かめよう
• （あなた自身の／あなたの）その目で確かめよう
• （あなた自身の／あなたの）その心で感じよう

The その in その手に触れるまで is very similar but it's used more as a simple "your" and it sounds like the speaker is thinking about a hand of someone specific like someone the speaker is in love with, etc. Unlike the examples above, the translation for this one would be until I touch your hand. It can't really be translated as until I touch your own hand.