TSLint's variable-name rule existed in gts@1:

Per typescript-eslint's ROADMAP style section (migration guide?), the TSLint variable-name equivalent is now @typescript/naming-convention. See rules/naming-convention.

Since variable-name was removed and naming-convention was not included, variable name linting is now more relaxed, i.e. a possible regression.

While the new naming-convention allows for significantly more control, a 1:1 replacement would be:

---
rules:
  '@typescript-eslint/naming-convention':
    - error
    - selector: variable
      format: ['camelCase', 'UPPER_CASE']
      leadingUnderscore: allow
      trailingUnderscore: allow
      custom:
        regex: 'any|[Nn]umber|[Ss]tring|[Bb]oolean|[Uu]ndefined'
        match: false

naming-convention does not explicitly include support for the ban-keywords option, so custom with a regex was included above to provide equivalency. See https://github.com/palantir/tslint/blob/8efa78e35a88b62775ce3c9bf0fa00586b0d34e1/src/rules/variableNameRule.ts#L26-L36