Fix docker_lite_push make target

[frouioui]

Description

In #16042 I modified the make targets touching our vitess/lite docker image, and the push target is a bit broken, it only pushes suffixed images (percona80, etc) and not latest. This PR is a merge of what we had before + what was fixed in 16042.

Backporting this to 20.0 as the original PR was merge there too.

[vitess-bot]

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 68.90%. Comparing base (639b1de) to head (0f0e65a).
Report is 3 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main   #16662      +/-   ##
==========================================
- Coverage   68.92%   68.90%   -0.02%     
==========================================
  Files        1562     1562              
  Lines      200855   200855              
==========================================
- Hits       138443   138405      -38     
- Misses      62412    62450      +38     

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