The AITiny button appears in your TinyMCE or JCE editor toolbar. It's the little colourful button with the small AI text inside it.
The button will only appear if the plugin is enabled and configured, and your user has access to at least one of the AITiny features. Please remember that there is a User Groups setting in the plugin. If your user account is not explicitly assigned to one of the assigned user groups you will not see the AI button.
Do keep in mind that each feature of the plugin has a toggle switch which can be used to display it. Moreover, custom actions also have their own user groups setting. If the combination of your options is such that no AITiny features would be displayed, you will not see the AI button at all; there's no point showing you a button that does nothing.
AITiny only supports Joomla's built-in TinyMCE editor (Editor - TinyMCE plugin) and the third-party JCE editor (Editor - JCE plugin). When it comes to JCE, it supports both the free of charge version, and the for-a-fee Professional version.
The AITiny button will appear wherever Joomla renders an editor using either TinyMCE or JCE. This includes both core Joomla and third party components.
For example, AITiny may appear in the Akeeba Ticket System reply area if you are using TinyMCE or JCE as your editor and there is at least one AITiny writing tool accessible to the user.
The exact location of the button depends on which editor you are using, and how you have configured it.
If you are using Joomla's built-in TinyMCE editor, the button will appear next to the Joomla drop-down button. That's the drop-down with all of the extra editor buttons (the published plugins in the editor-xtd group). If there is no such drop-down button—for example, if you have created a custom TinyMCE profile which does not have this button, or if there are no published plugins in the editor-xtd group either at all or available to your user— the AITiny button will be placed at the end of the editor toolbar. If your toolbar is multiple rows, it will be at the end of the last row.
AITiny works by injecting configuration options into the default profile of the TinyMCE editor at runtime. Unlike other TinyMCE plugins you do not have to edit the Editor - TinyMCE options yourself. The process used is similar to and builds on the way Joomla itself works. As such, it's generally safe when using caching. That is to say, Joomla's caching should not interfere with whether and to whom the plugin's features appear, as long as the edit page is either in the back-end of the site, or if it's a front-end page which correctly sets itself as not-cacheable (all built-in Joomla components handle that correctly).
That said, if you are using a caching proxy or CDN in front of your site you must make sure that it does not override Joomla's caching settings by caching the edit page. If this happens, you might not see the AI button in your editor, or the features of this plugin may appear to be broken.
If you are using the JCE editor, the button will appear next to the Joomla drop-down button. That's the drop-down with all of the extra editor buttons (the published plugins in the editor-xtd group). If there is no such drop-down button—for example, if you have created a custom JCE profile which does not have this button, or if there are no published plugins in the editor-xtd group either at all or available to your user— the AITiny button will be placed at the end of the first row of the editor's toolbar.
AITiny works by injecting configuration options to the currently used JCE profile at runtime. The notes about caching made with regards to the TinyMCE editor apply here as well.