Thank you for answering! I have run the update utility earlier, without remarks, but now I got this message:
The Intel® Driver Update Utility reports the following message:
A customized computer manufacturer driver is installed on your computer. The Intel® Driver Update Utility is not able to update the driver. Installing a generic Intel driver instead of the customized computer manufacturer driver may cause technical issues. Contact your computer manufacturer for the latest driver for your computer.
The words "may cause technical issues" seems a little dangerous.. How likeliy is it to run into problems if I download the driver (Version 184.108.40.20658) on this page:? Intel® Download Center
You can either download one from the manufacturer site: http://h20565.www2.hp.com/portal/site/hpsc/template.PAGE/public/psi/swdHome/?sp4ts.oid=5271147&spf_p.tpst=swdMain&spf_p.prp_swdMain=wsrp-navigationalState%3DswEnvOID%253D4132%257CswLang%253D%257Caction%253DlistDriver&javax.portlet.begCacheTok=com.vignette.cachetoken&javax.portlet.endCacheTok=com.vignette.cachetoken#Driver - Graphics
I see they have an updated version released not so long ago
Or You can download the latest one from the Intel® Download Center as You have found (it's also the correct one since this model of Yours have no additional graphics cards)
I took the middle way: Searched at the HP website for their latest driver, and installed that, version 10.18.10.3621. It works, lovely!