I suspect that you are inserting the link by going to Word's Insert tab, Text group, then clicking the Object dropdown and selecting Object. If you go that route, Word does not offer any of the internal objects in the Visio file.
If so, you may find it easier to start with the Visio file open in Visio (one reason is that you don't have to remember the names of all the pages/objects), then
- Select the tab of the page that you want to link to
- Click Copy
- In Word's Home tab, Click the arrow at the bottom of the Paste button
- Click Paste Special...
- Click the Paste Link radio button
- Click the format you want (e.g. Microsoft Visio Drawing object)
- Click OK
(a) you can link to things other than complete pages such as individual shapes and grouped shapes using this technique (or the one described by Johnny Estilles, as long as you know the correct name to use for the shape).
(b) I think in Visio you will always get the first page in the closed file on disk when you insert the link using the Insert->Object route, even when Visio is open. If you resequence the page tabs in Visio, and update the link, nothing happens until you save and close. I mention this because not all programs behave the same way. e.g. when you do the same thing with an Excel workbook, if the workbook is open when you perform the insertion or update the link, what you get is the open Worksheet. If the workbook is closed, you get the first worksheet.