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For different numberings for Theorem, Lemma and Definitions, I referred here. It states that

Load beamer with the notheorems option and declare all theorem types "by hand" without introducing a common numbering scheme. (Note: This is somewhat a hack, as it breaks the countsect package option [theorem numbering by section] for every type but theorem.)

\documentclass[notheorems]{beamer}

\setbeamertemplate{theorem}[ams style]
\setbeamertemplate{theorems}[numbered]

\makeatletter
    \ifbeamer@countsect
      \newtheorem{theorem}{\translate{Theorem}}[section]
    \else
      \newtheorem{theorem}{\translate{Theorem}}
    \fi
    \newtheorem{corollary}{\translate{Corollary}}
    \newtheorem{fact}{\translate{Fact}}
    \newtheorem{lemma}{\translate{Lemma}}
    \newtheorem{problem}{\translate{Problem}}
    \newtheorem{solution}{\translate{Solution}}

    \theoremstyle{definition}
    \newtheorem{definition}{\translate{Definition}}
    \newtheorem{definitions}{\translate{Definitions}}

    \theoremstyle{example}
    \newtheorem{example}{\translate{Example}}
    \newtheorem{examples}{\translate{Examples}}


    % Compatibility
    \newtheorem{Beispiel}{Beispiel}
    \newtheorem{Beispiele}{Beispiele}
    \theoremstyle{plain}
    \newtheorem{Loesung}{L\"osung}
    \newtheorem{Satz}{Satz}
    \newtheorem{Folgerung}{Folgerung}
    \newtheorem{Fakt}{Fakt}
    \newenvironment{Beweis}{\begin{proof}[Beweis.]}{\end{proof}}
    \newenvironment{Lemma}{\begin{lemma}}{\end{lemma}}
    \newenvironment{Proof}{\begin{proof}}{\end{proof}}
    \newenvironment{Theorem}{\begin{theorem}}{\end{theorem}}
    \newenvironment{Problem}{\begin{problem}}{\end{problem}}
    \newenvironment{Corollary}{\begin{corollary}}{\end{corollary}}
    \newenvironment{Example}{\begin{example}}{\end{example}}
    \newenvironment{Examples}{\begin{examples}}{\end{examples}}
    \newenvironment{Definition}{\begin{definition}}{\end{definition}}
\makeatother

\begin{document}

\begin{frame}

\begin{theorem}[A theorem]
Some text.
\end{theorem}

\begin{definition}[A definition]
Some text.
\end{definition}

\end{frame}

\end{document}

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The issue I am facing is while using \label{eq:abc} and \ref{eq:abc}. In the output the reference is showing the incorrect numberings. I looked into the matter and it seems that it is numbering as number of theorems+definitions+lemma+1.

How do I get the correct output?

2

Use shared counters, as described in chapter 3 of the amsthm manual.

Here, I’ve taken my best ansatz about which environments you want to share numbering.

\documentclass[notheorems]{beamer}

\setbeamertemplate{theorem}[ams style]
\setbeamertemplate{theorems}[numbered]

\makeatletter
    \newcounter{theoremcnt}
    \ifbeamer@countsect
      \newtheorem{theorem}[theoremcnt]{\translate{Theorem}}[section]
    \else
      \newtheorem{theorem}[theoremcnt]{\translate{Theorem}}
    \fi
    \newcounter{corollarycnt}
    \newtheorem{corollary}[corollarycnt]{\translate{Corollary}}
    \newcounter{factcnt}
    \newtheorem{fact}[factcnt]{\translate{Fact}}
    \newcounter{lemmacnt}
    \newtheorem{lemma}[lemmacnt]{\translate{Lemma}}
    \newcounter{problemcnt}
    \newtheorem{problem}[problemcnt]{\translate{Problem}}
    \newcounter{solutioncnt}
    \newtheorem{solution}[solutioncnt]{\translate{Solution}}

    \theoremstyle{definition}
    \newcounter{definitioncnt}
    \newtheorem{definition}[definitioncnt]{\translate{Definition}}
    \newtheorem{definitions}[definitioncnt]{\translate{Definitions}}

    \theoremstyle{example}
    \newcounter{examplecnt}
    \newtheorem{example}[examplecnt]{\translate{Example}}
    \newtheorem{examples}[examplecnt]{\translate{Examples}}


    % Compatibility
    \newtheorem{Beispiel}[examplecnt]{Beispiel}
    \newtheorem{Beispiele}[examplecnt]{Beispiele}
    \theoremstyle{plain}
    \newtheorem{Loesung}[solutioncnt]{L\"osung}
    \newtheorem{Satz}[theoremcnt]{Satz}
    \newtheorem{Folgerung}[lemmacnt]{Folgerung}
    \newtheorem{Fakt}[factcnt]{Fakt}
    \newenvironment{Beweis}{\begin{proof}[Beweis.]}{\end{proof}}
    \newenvironment{Lemma}{\begin{lemma}}{\end{lemma}}
    \newenvironment{Proof}{\begin{proof}}{\end{proof}}
    \newenvironment{Theorem}{\begin{theorem}}{\end{theorem}}
    \newenvironment{Problem}{\begin{problem}}{\end{problem}}
    \newenvironment{Corollary}{\begin{corollary}}{\end{corollary}}
    \newenvironment{Example}{\begin{example}}{\end{example}}
    \newenvironment{Examples}{\begin{examples}}{\end{examples}}
    \newenvironment{Definition}{\begin{definition}}{\end{definition}}
\makeatother

\begin{document}

\begin{frame}

\begin{theorem}[A theorem]\label{thm:1}
Some text.
\end{theorem}

\begin{definition}[A definition]\label{def:1}
Some text.
\end{definition}

Theorem \ref{thm:1} follows directly from Definition \ref{def:1}.

\end{frame}

\end{document}

Is this a Beispiel?

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