Try the different numbering schemes. It seems that some just continue the numbers on a single levels, while others use the expected multilevel numbering.

(I am still trying with Word 2003, and I didn't try with styles yet; only assigning numbering directly.)

>>Under the menu command "Format - Numbering...", you can select a multi-level numbering scheme. I guess you could include this as part of the style definition. However, the numbers are placed to the left, not to the right as in your example.
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>Yes, you are right. It should be placed on the left.
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>>You have more flexibility - and more work - if you use auto-numbering fields. It seems the appropriate field, in this case, is LISTNUM. Unfortunately, you have to manually insert it into each and every header. More information on LISTNUM here: http://support.microsoft.com/kb/162895/en-us/
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>By using the style, I was able to find the numbering options. This works for my Header1 style. For every chapter, I have 1, 2, 3 and so on. But, I haven't found an option would allow me to have the sub styles to include the parent representation in its number. For example, if I have two sections for Header2 under Header1, they will be 1, 2, 3 and so on no matter if I change chapter (by the use of Header1). So, they will not restart at 1. And, as mentioned, they do not include the parent representation.
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>So, I end up with:
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>

>1. This is title for Header1 style
>   1.  This is title for Header2 style
>   2. This is title for Header2 style
>2. This is title for Header1 style
>   1.  This is title for Header2 style
>   2. This is title for Header2 style
>3. This is title for Header1 style
>4. This is title for Header1 style
>

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>instead of:
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>1. This is title for Header1 style
>   1.1.  This is title for Header2 style
>   1.2. This is title for Header2 style
>2. This is title for Header1 style
>   2.1.  This is title for Header2 style
>   2.2. This is title for Header2 style
>3. This is title for Header1 style
>4. This is title for Header1 style
>