The ESV quoted by the OP is a perfectly valid translation, reasonably faithful and literal rendering of the Hebrew in both cases.
I would understand both these in a quote literal sense, just like in Deuteronomy 20:4 (NASB)
for the Lord your God is the One who is going with you, to fight for you against your enemies, to save you.’
This remains true whether they could actually see God or not, but that does not alter the fact of what these verses unambiguously say.
The holiness of the camp of Israel, the people, etc, is part of the general theme of the whole Torah as seen by the following examples:
- Ex 3:4-7, Moses instructed to remove his sandals in the presence of the LORD
- Ex 19:4, 5, The Israelites, at the inauguration of the Covenant of Israel were told that they were to be a "holy nation".
- Lev 11:44, 45, Israelites instructed to "be holy because I [the LORD] am holy".
- Lev 19:2 - Speak to the whole congregation of Israel and tell them: Be holy because I, the LORD your God, am holy.
- Lev 20:8 - Keep my decrees and follow them. I am the LORD, who makes you holy
- Lev 22:32 - I am the LORD who makes you holy
- Ex 31:13 - This will be a sign between me and you for generations to come so
that you will know that I am the LORD who makes you holy.
- Lev 21:23 - I am the LORD who makes them holy.
- Lev 22:16 - I am the LORD who makes them holy.
- Lev 22:9 - I am the LORD who makes them holy.
- Deut 23:14 - the camp is Israel must be holy because of the presence of the LORD.
Even after the Torah, the same principle still applied. For example:
- Josh 5:13-15 - When the "commander of the LORD's army" shows up, Joshua did two things (a) fell on his face to the ground in worship, and, (b) removed his sandals because the ground where he stood was holy
- Eze 37:28 - ... the nations will know that I am the LORD, who makes Israel holy.
Even in the NT this theme continues as referenced by 1 Peter several times:
- 1 Peter 1:15, 16 - But just as He who called you is holy, so be holy in all you do, for it is written: “Be holy, because I am holy.
- 1 Peter 2:9 - But you are a chosen people, a royal priesthood, a holy nation, a people for God’s own possession, to proclaim the virtues of Him who called you out of darkness into His marvelous light.
- 1 Peter 12 - Conduct yourselves with such honor among the Gentiles that, though they slander you as evildoers, they may see your good deeds and glorify God on the day He visits us. See also Matt 5:16.
Thus, the covenant people, both ancient and modern must be holy people in all that they do to reflect the holy God whom they serve and behave as if the LORD walks among them.