Similarly to Custom type disappears in subsequent citations you need to branch the code for cite:short to obtain a different format for @online entries. A short version of the title can be specified in the shorttitle field. But that would require manual intervention, with Biber's sourcemapping you can automate that. I strongly suggest to scrutinise the output carefully in that case since the code is quite naive and chops off everything at the third word.
The MWE builds on the code from Move date before title in bibliography using biblatex to make the output for the full citation match the style of the short citation.
\documentclass{article}
\usepackage[style=verbose, autocite=footnote]{biblatex}
\DeclareFieldFormat[online]{shorttitle}{\mkbibquote{#1\dots\isdot}}
\newbibmacro*{cite:short:online}{%
\printnames{labelname}%
\setunit{\addcomma\space}%
\printlist{organization}%
\setunit{\addcomma\space}%
\printdate
\setunit{\addcomma\space}%
\printtext[bibhyperlink]{%
\iffieldundef{shorttitle}
{\printfield[citetitle]{labeltitle}}
{\printfield{shorttitle}}}}
% cite:short is really 'cite:subsequent'
\renewbibmacro*{cite:short}{%
\ifbibmacroundef{cite:short:\strfield{entrytype}}
{\printnames{labelname}%
\setunit*{\printdelim{nametitledelim}}%
\printtext[bibhyperlink]{%
\printfield[citetitle]{labeltitle}}}
{\usebibmacro*{cite:short:\strfield{entrytype}}}}
\DeclareSourcemap{
\maps[datatype=bibtex]{
\map[overwrite=false]{
\step[fieldsource=title, match=\regexp{\A(\w+\s+){2}(\w+)}, final]
\step[fieldset=shorttitle, origfieldval, final]
\step[fieldsource=shorttitle, match=\regexp{\A(\w+\s+)(\w+\s+)(\w+).*}, replace={$1$2$3}]
}
}
}
\DefineBibliographyStrings{english}{
urlseen = {visited at}
}
\DeclareFieldFormat{urldate}{\bibstring{urlseen}\space#1}
\DeclareListWrapperFormat{organization}{\mkbibemph{#1}}
\DeclareBibliographyDriver{online}{%
\usebibmacro{bibindex}%
\usebibmacro{begentry}%
\usebibmacro{author/editor+others/translator+others}%
\newunit\newblock
\printlist{organization}%
\newunit\newblock
\usebibmacro{date}%
\newunit\newblock
\usebibmacro{title}%
\newunit
\printlist{language}%
\newunit\newblock
\usebibmacro{byauthor}%
\newunit\newblock
\usebibmacro{byeditor+others}%
\newunit\newblock
\printfield{version}%
\newunit
\printfield{note}%
\newunit\newblock
\iftoggle{bbx:eprint}
{\usebibmacro{eprint}}
{}%
\newunit\newblock
\usebibmacro{url+urldate}%
\newunit\newblock
\usebibmacro{addendum+pubstate}%
\setunit{\bibpagerefpunct}\newblock
\usebibmacro{pageref}%
\newunit\newblock
\iftoggle{bbx:related}
{\usebibmacro{related:init}%
\usebibmacro{related}}
{}%
\usebibmacro{finentry}}
\usepackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@online{gates,
author = {Bill Gates},
title = {Save the world!},
url = {https://www.gatesfoundation.org/de/},
urldate = {2016-07-04},
organization = {{Bill and Melinda Gates Foundation}},
}
@online{gates2,
author = {Bill Gates},
title = {Save the moon with even longer titles},
shorttitle = {Save the moon},
date = {2016-05-04},
url = {https://www.gatesfoundation.org/de/},
urldate = {2016-07-04},
organization = {Bill {and} Melinda Gates Foundation},
}
@online{ex:one,
author = {Anne Elk},
title = {One},
url = {https://example.com/~elk/1},
}
@online{ex:two,
author = {Anne Elk},
title = {Two Words},
url = {https://example.com/~elk/2},
}
@online{ex:three,
author = {Anne Elk},
title = {Three Words Title},
url = {https://example.com/~elk/3},
}
@online{ex:four,
author = {Anne Elk},
title = {Four Words in Title},
url = {https://example.com/~elk/4},
}
@online{ex:five,
author = {Anne Elk},
title = {Five Words in a Title},
url = {https://example.com/~elk/5},
}
@online{ex:long,
author = {Anne Elk},
title = {This Title Is Very Long to Test the Feature},
url = {https://example.com/~elk/long},
}
@online{ex:override,
author = {Anne Elk},
title = {A Long Title That Will Be Overridden},
shorttitle = {A Different Title},
url = {https://example.com/~elk/over},
}
\end{filecontents*}
\addbibresource{\jobname.bib}
\begin{document}
Lorem ipsum dolor sit amet, consetetur sadipscing elitr,\autocite{gates} sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. \autocite{gates2}
Lorem ipsum dolor sit amet, consetetur sadipscing elitr,\autocite{gates} sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. \autocite{gates2}
Lorem\autocite{ex:one} ipsum\autocite{ex:one}.
Lorem\autocite{ex:two} ipsum\autocite{ex:two}.
Lorem\autocite{ex:three} ipsum\autocite{ex:three}.
Lorem\autocite{ex:four} ipsum\autocite{ex:four}.
Lorem\autocite{ex:five} ipsum\autocite{ex:five}.
Lorem\autocite{ex:override} ipsum\autocite{ex:override}.
Lorem\autocite{ex:long} ipsum\autocite{ex:long}.
\printbibliography
\end{document}

In the match–replace RegExp I went with the naive
match=\regexp{\A(\w+\s+)(\w+\s+)(\w+).*}, replace={$1$2$3}
if you want a more sophisticated solution that can be extended more easily, you could try
match=\regexp{\A((?:\w+\s+){2})(\w+).*}, replace={$1$2}
where ((?:\w+\s+){2}) captures the first 2 words and the (\w+) the following word.
@online? I also feel that in this case the "..." is pretty useless, why not just include the "!" to get the full title? – moewe Dec 22 '18 at 14:25@online. You're right, in this exemple it doesn't make that much sense, but with longer titles it does make sense. – czer Dec 22 '18 at 14:29urldatebetweentitleandorganizationhere? In https://tex.stackexchange.com/q/464205/35864 that place is taken by the normal publication datedate. – moewe Dec 22 '18 at 14:38datefield, my answer should already have you covered. I assumed you did not want theurldate. I suggest you add thedateto your example entry so the question makes more sense. – moewe Dec 22 '18 at 16:22andin the name of theorganizationyou should use double braces, eitherorganization = {{Bill and Melinda Gates Foundation}},ororganization = {Bill {and} Melinda Gates Foundation},(I mentioned that before in https://tex.stackexchange.com/a/464276/35864). – moewe Dec 22 '18 at 16:23ext-authortitle-ibid. I'm really sorry for that. At the point I posted the question I didn't knew that I will need to useext-authortitle-ibidinstead ofverbose– czer Dec 22 '18 at 16:55verboseandext-authortitle-ibid(or justauthortitle-ibid) give completely different output on the first citation. – moewe Dec 22 '18 at 16:56style=ext-authortitle, citestyle=verbose-ibidworks perfectly for me. Probably there are some disadvantages to that approach but so far it works for me. – czer Dec 22 '18 at 18:07style=ext-verbose-ibidwould be more natural and should give the same results. – moewe Dec 22 '18 at 19:47