[Bug]: Integrated terminal does not set TERM_PROGRAM, breaking shell prompt and terminal detection

Before submitting

• I searched existing issues and did not find a duplicate.
• I included enough detail to reproduce or investigate the problem.

Area

apps/desktop

Steps to reproduce

1. Open the integrated terminal in T3 Code on macOS.
2. Run: echo "[$TERM_PROGRAM] [$TERM_PROGRAM_VERSION] [$COLORTERM]"
3. Observe all three are empty.

For a practical repro, use any shell config that selects a prompt by terminal.
Minimal case:

case "$TERM_PROGRAM" in
  "vscode")   echo "vscode config" ;;
  "ghostty")  echo "ghostty config" ;;
  *)          echo "fallback config" ;;
esac

In every other terminal emulator this selects the correct branch.
In T3 Code it always hits the fallback, because there is nothing to match on.

Expected behavior

The integrated terminal exports TERM_PROGRAM (and ideally TERM_PROGRAM_VERSION)
into the pty environment, the way every other terminal emulator does:

Emulator	TERM_PROGRAM
VS Code	vscode
Apple Terminal	Apple_Terminal
iTerm2	iTerm.app
Ghostty	ghostty
WezTerm	WezTerm
Hyper	Hyper
T3 Code	(unset)

Something like TERM_PROGRAM=t3code plus TERM_PROGRAM_VERSION=0.0.35.

Actual behavior

TERM_PROGRAM is never set, so the shell cannot tell it is running inside T3 Code.

Impact

Minor bug or occasional failure

Version or commit

main

Environment

15.7.4

Logs or stack traces

Screenshots, recordings, or supporting files

No response

Workaround

No response

The affected area is apps/desktop's integrated terminal and its pty environment; start by tracing where the terminal process environment is constructed. Verify that TERM_PROGRAM and TERM_PROGRAM_VERSION are present with the intended values by repeating the issue's echo command on macOS, and confirm shell prompt and terminal detection select the T3 Code branch.