I guess that I ought to put the direct URLs here as well for the latest 7.10.x and 7.11.x versions of the software. The .htaccess file relies on mod_rewrite (in apache) to carry out the following URL changes:
RewriteRule ^Api/(.*)$ - [env=HTTP_AUTHORIZATION:%{HTTP:Authorization}]
RewriteRule ^Api/access_token$ Api/index.php/access_token [L]
RewriteRule ^Api/V8/(.*?)$ Api/index.php/V8/$1 [L]
This means that you can avoid the shortened https://mysite.com/Api/access_token and instead use the full path https://mysite.com/Api/index.php/access_token
Similarly for using the V8 Api, replace https://mysite.com/Api/V8/module and instead use the full path https://mysite.com/Api/index.php/V8/module
This way, you should still get the desired REST connection, even if the rewrite rules/.htaccess etc is messed up in some way.